Download Building a mechanistic model of the development and function of the

Journal of Physiology - Paris xxx (2012) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Journal of Physiology - Paris
journal homepage: www.elsevier.com/locate/jphysparis
Building a mechanistic model of the development and function of the primary
visual cortex
James A. Bednar
Institute for Adaptive and Neural Computation, The University of Edinburgh, 10 Crichton St., EH8 9AB Edinburgh, UK
a r t i c l e
i n f o
Article history:
Available online xxxx
Keywords:
Topographic map
Development
Computational model
Orientation map
Visual system
Cortical microcircuit
Cortical column
a b s t r a c t
Researchers have used a very wide range of different experimental and theoretical approaches to help
understand mammalian visual systems. These approaches tend to have quite different assumptions,
strengths, and weaknesses. Computational models of the visual cortex, in particular, have typically implemented either a proposed circuit for part of the visual cortex of the adult, assuming a very specific wiring
pattern based on findings from adults, or else attempted to explain the long-term development of a visual
cortex region from an initially undifferentiated starting point. Previous models of adult V1 have been able
to account for many of the measured properties of V1 neurons, while not explaining how these properties
arise or why neurons have those properties in particular. Previous developmental models have been able
to reproduce the overall organization of specific feature maps in V1, such as orientation maps, but are
generally formulated at an abstract level that does not allow testing with real images or analysis of
detailed neural properties relevant for visual function. In this review of results from a large set of new,
integrative models developed from shared principles and a set of shared software components, I show
how these models now represent a single, consistent explanation for a wide body of experimental evidence, and form a compact hypothesis for much of the development and behavior of neurons in the visual
cortex. The models are the first developmental models with wiring consistent with V1, the first to have
realistic behavior with respect to visual contrast, and the first to include all of the demonstrated visual
feature dimensions. The goal is to have a comprehensive explanation for why V1 is wired as it is in
the adult, and how that circuitry leads to the observed behavior of the neurons during visual tasks.
Ó 2012 Published by Elsevier Ltd.
1. Introduction
Understanding how we see remains an elusive goal, despite
more than half a century of intensive work using a wide array of
experimental and theoretical techniques. Because each of these
techniques has different assumptions, strengths, and weaknesses,
it can be difficult to establish clear principles and conclusive evidence. To make significant progress in this area, it is important
to consider how the existing data can be synthesized into a coherent explanation for a wide variety of phenomena.
Computational models of the primary visual cortex (V1) could
provide a platform for achieving such a synthesis, integrating results across levels to provide an overall explanation for the main
body of results. However, existing models typically fall into one
of two categories with different aims, neither of which achieves
this goal: (1) narrowly constrained models of specific aspects of
adult cortical circuitry or function, or (2) abstract models of
E-mail address: [email protected]
URLs: http://homepages.inf.ed.ac.uk/jbednar
large-scale visual area development, accounting for only a few of
the response properties of individual neurons within these areas.
Existing models of type (1) (e.g. Chance et al., 1999; Adelson
and Bergen, 1985; Albrecht and Geisler, 1991) have been able to
show how a variety of specialized circuits (often mutually incompatible) can account for most of the major observed functional
properties of V1 neurons, but do not attempt to show how a single,
general-purpose circuit could explain most of them at the same
time. Because each specific phenomenon can often be explained
by many different specialized models, it can be difficult or impossible to distinguish between different explanations. Moreover, just
showing an example of how the property can be implemented does
little to explain why neurons are arranged in this way, and what
the circuit might contribute to the process of vision.
Similarly, many existing models of type (2) have been able to account for the large-scale organization of V1, such as its arrangement
into topographic orientation maps (reviewed in Swindale (1996),
Goodhill (2007), and Erwin et al. (1995)). Yet because the developmental models are formulated at an abstract level, they address only
a few of the observed properties of V1 neurons (such as their orientation or eye preference), and it is again difficult to decide between
0928-4257/$ - see front matter Ó 2012 Published by Elsevier Ltd.
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
2
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
the various explanations. Thus despite the many thousands of experimental and computational papers about the visual cortex, methods
for integrating, interpreting, and evaluating the overall body of evidence to build a coherent explanation remain frustratingly scarce.
This paper outlines and reviews a large set of closely interrelated computational models of the visual cortex that together are
beginning to form a consistent, biologically grounded, computationally simple explanation of the bulk of V1 development and
function. Specifically, the models develop:
1. Neurons with receptive fields (RFs) selective for retinotopy
(X, Y), orientation (OR), ocular dominance (OD), motion direction (DR), spatial frequency (SF), temporal frequency (TF), disparity (DY), and color (CR).1
2. Preferences for each of these organized into realistic spatially
topographic maps.
3. Lateral connections between these neurons that reflect the
structure of the maps.
4. Realistic surround modulation effects, including their diversity,
caused by interactions between these neurons.
5. Contrast-gain control and contrast-invariant tuning for the individual neurons, ensuring that they retain selectivity robustly.
6. Both simple and complex cells, to account for the major
response types of V1 neurons.
7. Long-term and short-term plasticity (e.g. aftereffects), emerging
from mechanisms originally implemented for development.
Together, these phenomena arguably represent the bulk of the
generally agreed stimulus-driven response properties of V1 neurons. Accounting for such a diverse set of phenomena could have
required an extremely complex model, e.g. the union of the many
previously proposed models for each of the individual phenomena.
Yet our results show that it is possible to account for all of these
using only a small set of plausible principles and mechanisms,
within a consistent biologically grounded framework:
1. Single-compartment (point) firing-rate (non-spiking) RGC, LGN,
and V1 neurons.
2. Hardwired subcortical pathways to V1 including the main LGN
(lateral geniculate nucleus) or RGC (retinal ganglion cell) types
that have been identified.
3. Initially isotropic, topographic connectivity within and between
neurons in layers in V1.
4. Natural images and spontaneous activity patterns that lead to
V1 responses.
5. Hebbian learning with normalization for V1 neurons.
6. A large number of parameters associated with each of these
mechanisms.
Properties not necessary to explain the phenomena above, such
as spiking and detailed neuronal morphology, have been omitted,
to clearly focus on the most relevant aspects of the system. The
overall hypothesis is that much of the complex structure and properties observed in the visual cortex emerges from interactions between relatively simple but highly interconnected computing
elements, with connection strengths and patterns self-organizing
in response to visual input and other sources of neural activity.
1
Abbreviations: OR: orientation, OD: ocular dominance, DR: motion direction, SF:
spatial frequency, TF: temporal frequency, DY: disparity, CR: color, RF: receptive field,
CF: connection field, RGC: retinal ganglion cell, LGN: lateral geniculate nucleus, V1:
primary visual cortex, GCAL: gain-control, adaptive laterally connected (model),
LISSOM: Laterally interconnected synergetically self-organizing map, LMS: long,
medium, short (wavelength cone photoreceptors), GR: medium-center, long-surround retinal ganglion cell, RG: long-center, medium-surround retinal ganglion cell,
BY: short-center, long and medium surround retinal ganglion cell, OCTC: orientationcontrast tuning curve.
Through visual experience, the geometry and statistical regularities of the visual world become encoded into the structure and
connectivity of the visual cortex, leading to a complex functional
cortical architecture that reflects the physical and statistical properties of the visual world.
At present, many of the results have been obtained independently in a wide variety of separate projects performed by different
collaborators at different times. However, all of the models share
the same underlying principles outlined above, and all are implemented using the same simulator and a small number of underlying components. This review shows how each of these modeling
studies, previously reported separately, fits into a consistent and
compact framework for explaining a very wide range of data. The
models are the first developmental models of V1 maps with wiring
consistent with V1, and the first to have realistic behavior with respect to visual contrast, and together they account for all of the
various spatial feature dimensions for which topographic maps
have been reported using imaging in mammals.
Preliminary work into developing an implementation combining each of the models into a single, working model visual system
is also reported, although doing so is still a long-term work in progress. The unified model will include all of the visual feature
dimensions, as well as all of the major sources of connectivity that
affect V1 neuron responses. The goal is to have the first comprehensive, mechanistic explanation for why V1 becomes wired as it
is in the adult, and how that circuitry leads to the observed behavior of the neurons during visual tasks. That is, the model will be the
first that starts from an initially undifferentiated state, to wire itself into a collection of neurons that behave, at a first approximation, like those in V1. Because such a model starts with no
specializations (at the cortical level) specific to vision and would
organize very differently when given different inputs, it would also
represent a general explanation for the development and function
of sensory and motor areas throughout the cortex.
2. Material and methods
All of the models whose results are presented here are implemented in the Topographica simulator, and are freely available
along with the simulator from www.topographica.org. This section describes the complete, unified architecture under development, with each currently implemented model representing a
subset or simplification of this architecture (as specified for each
set of results below). The proposed model is a generalization of
the GCAL (gain-controlled, adaptive, laterally connected) map
model (Law et al., 2011), extended to cover results from a large
family of related models. The unified GCAL model is intended to
represent the visual system of the macaque monkey, but relies
on data from studies of cats, ferrets, tree shrews, or other mammalian species where clear results are not yet available from
monkeys.
2.1. Sheets and projections
Each Topographica model consists of a set of sheets of neurons
and projections (sets of topographically mapped connections) between them. For each model discussed in this paper, there are
sheets representing the visual input (as a set of activations in photoreceptor cells), the transformation from the photoreceptors to inputs driving V1 (expressed as a set of LGN cell activations), and
neurons in V1. Fig. 1 shows the sheets and connections in the unified GCAL model, which is described for the first time here as a single, combined model.
Each sheet is implemented as a two-dimensional array of firingrate neurons. The Topographica simulator allows parameters for
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
3
tially localized set of connections from neurons in an input sheet
near the corresponding topographic location of the target neuron.
Fig. 1 shows one sample connection field (CF) for each projection,
visualized as an oval of the corresponding radius on the input sheet
(drawn to scale), connected by a cone to the neuron on the target
sheet. The connections and their weights determine the specific
properties of each neuron in the network, by differentially weighting RGC/LGN inputs of different types and/or locations. Each of the
specific types of sheets and projections is described in the following sections.
2.2. Images and photoreceptor sheets
Fig. 1. Comprehensive GCAL model architecture. Unified GCAL model for simple
and complex cells with surround modulation and (X, Y), OR, OD, DY, DR, TF, SF, and
CR maps (see list of abbreviations on page 4). The model consists of 29 neural sheets
and 123 separate projections between them. Each sheet is drawn to scale, with
larger sheets subcortically to avoid edge effects, and an actual sample activity
pattern on each subcortical sheet. Each projection is illustrated with an oval, also
drawn to scale, showing the extent of the connection field in that projection, with
lines converging on the target of the projection. Sheets below V1 are hardwired to
cover the range of response types found in the retina and LGN. Connections to V1
neurons adapt via Hebbian learning, allowing initially unselective V1 neurons to
exhibit the range of response types seen experimentally, by differentially weighting
each of the subcortical and intracortical inputs.
sheets and projections to be specified in measurement units that
are independent of the specific grid sizes used in a particular run
of the simulation. To achieve this, Topographica sheets provide
multiple spatial coordinate systems, called sheet and matrix coordinates. Where possible, parameters (e.g. sheet dimensions or connection radii) are expressed in sheet coordinates, expressed as if
the sheet were a continuous neural field rather than a finite grid.
In practice, of course, sheets are always implemented using some
finite matrix of units. Each sheet has a parameter called its density,
which specifies how many units (matrix elements) in the matrix
correspond to a length of 1.0 in continuous sheet coordinates,
which allows conversion between sheet and matrix coordinates.
In all simulations shown, sheets of V1 neurons have dimensions
(in sheet coordinates) 1.0 1.0. If every other sheet in the model
were to have a 1.0 1.0 area, units near the border of higher-level
sheets like V1 would have afferent connections that extend past
the border of lower-level sheets like the RGC/LGN cells. This cropping of connections will result in artifacts in the behavior of units
near the border. To avoid such artifacts, lower-level sheets have
areas larger than 1.0 1.0. (Alternatively, one could avoid cropping of connections by imposing periodic boundary conditions,
but doing so would create further artifacts by combining unrelated
portions of the visual field into the connection fields of V1 neurons.) In Fig. 1 each sheet is plotted at the same scale in terms of
degrees of visual angle covered, and thus the photoreceptor and
RGC/LGN sheets appear larger. Sheet dimensions were chosen to
ensure that each unit in the receiving sheet has a complete set of
connections, where possible, minimizing edge effects in the RGC/
LGN and V1 (Miikkulainen et al., 2005). When sizes are scaled
appropriately (Bednar et al., 2004), results are independent of the
density used, except at very low densities or for simulations with
complex cells, where complexity increases with density (as described below). Larger areas can be simulated easily (Bednar
et al., 2004), but require more memory and simulation time.
A projection to an m m sheet of neurons consists of m2 separate connection fields, one per target neuron, each of which is a spa-
The unified GCAL model contains six input sheets, representing
the Long, Medium, and Short (LMS) wavelength cone photoreceptors in the retinas of the left and right eyes (illustrated in Fig. 1).
The density of photoreceptors is uniform across the sheets, because
only a relatively small portion of the visual field is being modeled,
but the non-uniform spacing from fovea to periphery can be added
for a model with a larger input sheet. Given a color image appearing in one eye, the estimated activations of the LMS sheets are calculated from the cone sensitivity functions for each photoreceptor
type (Stockman et al., 1999; Stockman and Sharpe, 2000) using calibrated color images (Olmos and Kingdom, 2004), following the
method described in De Paula (2007). Input image pairs (left, right)
were generated by choosing one image randomly from a database
of single calibrated images, selecting a random patch within the
image, a random nearly horizontal offset between patterns in each
eye (as described in Ramtohul (2006)), a random direction of motion translation with a fixed speed (described in Bednar and
Miikkulainen (2003)), and a random brightness difference between
the two eyes (described in Miikkulainen et al. (2005)). These modifications are intended as a simple model of motion and eye differences, to allow development of direction preference, ocular
dominance, and disparity maps, until suitable full-motion stereo
calibrated-color video datasets of natural scenes are available. Simulated retinal waves can also be used as inputs, to provide initial RF
and map structure before eye opening, but are not required for RF
or map development in the model (Bednar and Miikkulainen,
2004).
2.3. Subcortical sheets
The subcortical pathway from the photoreceptors to the thalamorecipient cells in V1 is represented as a set of hardwired subcortical cells with fixed connection fields (CFs) that determine the
response properties of each cell. These cells represent the complete
processing pathway to V1, including circuitry in the retina (including the retinal ganglion cells), optic nerve, lateral geniculate nucleus, and optic radiations to V1. Because the focus of the model
is to explain cortical development given its thalamic input, the
properties of these RGC/LGN cells are kept fixed throughout development, for simplicity and clarity of analysis.
Each distinct RGC/LGN cell type is grouped into a separate
sheet, each of which contains a topographically organized set of
cells with identical properties but responding to a different region
of the retinal photoreceptor input sheet. Fig. 1 shows examples of
each of the different response types suitable for the development
of the full range of V1 RFs: SF1 (achromatic cells with large receptive fields, as in the magnocellular pathway), SF2 (achromatic cells
with small receptive fields), BY cells (color opponent cells with
blue (short) cone photoreceptor center, and red (long) and green
(short) surround), and similarly for medium-center (GR) and
long-center (RG) chromatic L/M RFs. Each such opponent cell
comes in two types, On (with an excitatory center) and Off (with
an excitatory surround).
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
4
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
All of these cells have Difference-of-Gaussian RFs, and thus perform edge enhancement at a particular size scale (SF1 and SF2)
and/or color enhancement (e.g. RG and GR). Each of these RF types
has been reported in the macaque retina or LGN (Vidyasagar et al.,
2002; Dacey, 2000; Klug et al., 2003), and together they cover the
range of typically reported spatial response types. Additional such
cell classes can easily be added as needed, e.g. to provide more
than two sizes of RGC/LGN cells with different spatial frequency
preferences (Palmer, 2009), although doing so increases memory
and computation requirements. Not all of the cell types currently
included are necessarily required for these results, but so far they
have been found to be sufficient.
For each of the RGC/LGN sheets, multiple projections with different delays connect them to the V1 sheet. These delays represent
the different latencies in the lagged vs. non-lagged cells found in
cat LGN (Saul and Humphrey, 1992; Wolfe and Palmer, 1998),
and allow V1 neurons to become selective for the direction of motion. Lagged cells could instead be implemented using separate
LGN sheets, as in Bednar and Miikkulainen (2003), if differences
from non-lagged cells other than temporal delay need to be taken
into account. Many other sources of temporal delays would also
lead to direction preferences, but have not been tested specifically.
Apart from these delays, the detailed temporal properties of the
subcortical neural responses and of signal propagation along the
various types of connections elsewhere in the network have not
been modeled. Instead, the model RGC/LGN neurons have a constant, sustained output, and all connections in each projection have
a constant delay, independent of the physical length of that connection. Modeling the subcortical temporal response properties
and simulating non-uniform delays would greatly increase the
number of timesteps needed to simulate each input presentation,
but should otherwise be feasible in future work.
2.4. Cortical sheets
Many of the simulations use only a single V1 sheet for simplicity, but in the full unified model, V1 is represented by three cortical
sheets. First, cells with direct thalamic input are labeled V1 L4 in
Fig. 1, and nominally correspond to pyramidal simple cells in macaque V1 layer 4Cb. Second, pyramidal cells in layer 2/3 (labeled V1
L2/3E) receive input from a small topographically corresponding
(columnar) region of V1 L4. These cells (as discussed below) become complex-cell–like, i.e. relatively insensitive to spatial phase,
by pooling across nearby L4 simple cells of different preferred spatial phases (Antolik and Bednar, 2011). The third V1 sheet L2/3I
models inhibitory interneurons in layer 2/3. Together, these sheets
will be used to show how simple cells can develop in V1 L4, how
complex cells can develop in L2/3, and how interactions between
these cells can vary depending on contrast, which affects the balance between excitation and inhibition (Antolik, 2010).
The behavior of the V1 sheets is primarily determined by the
projections to, within, and between them. Fig. 2 illustrates and describes each of these projections. The overall pattern of projections
was chosen based on anatomical tracing of the V1 microcircuit in
cat (Binzegger et al., 2009), providing the first model of V1 selforganization where the long-range connections are excitatory
(Law, 2009; Antolik, 2010), as found in animals (Gilbert and Wiesel, 1989). Each of these projections is initially non-specific, and
becomes selective only through the process of self-organization
(described below), which increases some connection weights at
the expense of others.
2.5. Activation
At each training iteration, a new retinal input image is presented and the activation of each unit in each sheet is updated in
Fig. 2. Projections in model V1. Each cone represents a projection to the neuron to
which it points. V1 L4 neurons receive numerous direct projections from the RGC/
LGN cells shown in Fig. 1 (green projections along the bottom of the figure), along
with weak short-range lateral excitatory connections (blue ovals) and feedback
connections from V1 L2/3E (light blue and red cones pointing to V1 L4). V1 L2/3E
neurons receive narrow afferent projections from V1 L4 (red cone on left), narrow
inhibitory projections from V1 L2/3I (red cone in top right), and both short-range
and long-range lateral excitatory projections (blue ovals). V1 L2/3I neurons receive
both short-range and long-range afferent excitation from V1 L2/3E (green and
purple cones), and make short-range lateral inhibitory connections to other
V1 L2/3I neurons (blue oval). If projections are visualized as outgoing rather than
incoming as drawn here, L2/3E cells will make wide-ranging excitatory connections
to cells of both L2/3E and L2/3I, and L2/3I cells will make short-ranging inhibitory
connections to cells of both sheets. This pattern of connectivity is a simplified but
consistent implementation of the known connectivity patterns between V1 layers
(Binzegger et al., 2009).
a series of steps. One training iteration represents one visual fixation (for natural images) or a snapshot of the relatively slowly
changing spatial pattern of spontaneous activity (e.g. for retinal
waves (Wong, 1999)). I.e., an iteration consists of a constant retinal
activation, followed by recurrent processing at the LGN and cortical levels. For one iteration, assume that input is drawn on the photoreceptors at time t and the connection delay (constant for all
projections) is defined as dt = 0.05 (roughly corresponding to 10–
20 ms). Then at t + 0.05 the RGC/LGN cells compute their responses, and at t + 0.10 the thalamic output is delivered to V1,
where it similarly propagates through the cortical sheets.
Images are presented to the model by activating the retinal
photoreceptor units. The activation value Wi,P of unit i in photoreceptor sheet P is given by the calibrated-color estimate of the L, M,
or S cone activation in the chosen image at that point.
For each model neuron in the other sheets, the activation value
is computed based on the combined activity contributions to that
neuron from each of the sheet’s incoming projections. The activity
contribution from a projection is recalculated whenever its input
sheet activity changes, after the corresponding connection delay.
For unit j in the target sheet, the activity contribution Cjp to j from
projection p is a dot product of the relevant input with the weights:
C jp ðt þ dtÞ ¼
P
X isp ðtÞxij;p
ð1Þ
i2F jp
where Xis is the activation of unit i on this projection’s input sheet
sp, taken from the set of all input neurons from which target unit
j receives connections in that projection (its connection field Fjp),
and xij,p is the connection weight from i to j in that projection.
Across all projections, multiple direct connections between the
same pair of neurons are possible, but each projection p contains
at most one connection between i and j, denoted by xij,p.
For a given subcortical or cortical unit j in the separate models
reported in this paper (except those containing complex cells), the
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
5
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
activity gj(t) is calculated from a rectified weighted sum of the
activity contributions Cjp(t):
gj ðtÞ ¼ f
P
p
!
cp C jp ðtÞ
ð2Þ
f is a half-wave rectifying function with a variable threshold point
(h) dependent on the average activity of the unit, as described in
the next subsection. Each cp is an arbitrary multiplier for the overall
strength of connections in projection p. The cp values are set in the
approximate range 0.5–3.0 for excitatory projections and 0.5 to
3.0 for inhibitory projections. For afferent connections, the cp value is chosen to map average V1 activation levels into the range 0–
1.0 by convention, for ease of interconnecting and analyzing multiple sheets. For lateral and feedback connections, the cp values are
then chosen to provide a balance between feedforward, lateral,
and feedback drive, and between excitation and inhibition; these
parameters are crucial for making the network operate in a useful
regime.
For the full unified model, RGC/LGN neuron activity is computed similarly to Eq. (2), except that they have a separate divisive
lateral inhibitory projection:
0P
gjL ðtÞ ¼ f @
p
cp C jp ðtÞ 1
cS C jS ðtÞ þ k
A
ð3Þ
where L stands for one of the RGC/LGN sheets. Projection S consists
of a set of isotropic Gaussian-shaped lateral inhibitory connections
(see Eq. (7), evaluated with u = 1), and p ranges over all the other
projections to that sheet. k is a small constant to make the output
well-defined for weak inputs. The divisive inhibition implements
the contrast gain control mechanisms found in RGC and LGN neurons (Antolik, 2010; Felisberti and Derrington, 1999; Bonin et al.,
2005; Alitto and Usrey, 2008).
For the unified model and individual models with complex cells,
cortical neuron activity is computed similarly to Eq. (2), except to
add firing-rate-fluctuation noise and exponential smoothing of the
recurrent dynamics:
gjV ðt þ dtÞ ¼ kf
P
p
!
cp C jp ðt þ dtÞ þ ð1 kÞgjV ðtÞ þ rn x
ð4Þ
where V stands for one of the cortical sheets, p ranges over all projections to that sheet, k = 0.5 is a time constant parameter that defines the strength of smoothing of the recurrent dynamics in the
network, x is a normally distributed zero-mean unit-variance random variable, and rn scales x to determine the amount of noise.
The smoothing ensures that the system remains numerically stable,
without spurious oscillations caused by simulating only discrete
time steps, for the relatively coarse time steps that are used here
for computational efficiency.
For any of the models, each time the activity is computed using
Eq. (2), (3), or (4), the new activity values are sent to each of the
outgoing projections, where they arrive after the projection delay
(typically 0.05). The process of activity computation then begins
again, with a new contribution C computed as in Eq. (1), leading
to new activation values by Eqs. (2), (3), or (4). Activity thus
spreads recurrently throughout the network, and can change, die
out, or be strengthened, depending on the parameters.
With typical parameters that lead to realistic topographic map
patterns, initial blurry patterns of afferent-driven activity are
sharpened into well-defined ‘‘activity bubbles’’ through locally
cooperative and more distantly competitive lateral interactions
(Miikkulainen et al., 2005). Nearby neurons are thus influenced
to respond more similarly, while more distant neurons receive
net inhibition and thus learn to respond to different input patterns.
The competitive interactions ‘‘sparsify’’ the cortical response into
patches, in a process that can be compared to the explicit sparseness constraints in non-mechanistic models (Olshausen and Field,
1996), while the local facilitatory interactions encourage spatial
locality so that smooth topographic maps will be developed.
Whenever the time reaches an integer multiple (e.g. 1.0 or 2.0),
the V1 response is used to update the threshold point (h) of V1
neurons (using the adaptation process described in the next section) and to update the afferent weights via Hebbian learning (as
described in the following section). Both adaptation and learning
could also be performed at each settling step, but doing so would
greatly decrease computational efficiency. Because the settling
(sparsification) process typically leaves only small patches of the
cortical neurons responding strongly, those neurons will be the
ones that learn the current input pattern, while other nearby neurons will learn other input patterns, eventually covering the complete range of typical input variation. Overall, through a
combination of the network dynamics that achieve sparsification
along with local similarity, plus Hebbian learning that leads to feature preferences, the network will learn smooth, topographic maps
with good coverage of the space of input patterns, thereby developing into a functioning system for processing patterns of visual
input.
2.6. Homeostatic adaptation
For this model, the threshold for activation of each neuron is a
very important quantity, because it directly determines how much
the neuron will fire in response to a given input. To set the threshold, each neuron unit j in V1 calculates a smoothed exponential
average of its activity (gj ):
gj ðtÞ ¼ ð1 bÞgj ðtÞ þ bgj ðt 1Þ
ð5Þ
The smoothing parameter (b = 0.999) determines the degree of
smoothing in the calculation of the average. gj is initialized to the
target average V1 unit activity (l), which for all simulations is
gjA ð0Þ ¼ l ¼ 0:024. The threshold is updated as follows:
hðtÞ ¼ hðt 1Þ þ kðgj ðtÞ lÞ
ð6Þ
where k = 0.0001 is the homeostatic learning rate. The effect of this
scaling mechanism is to bring the average activity of each V1 unit
closer to the specified target. If the activity in a V1 unit moves away
from the target during training, the threshold for activation is thus
automatically raised or lowered in order to bring it closer to the target. Note that an alternative rule with only a single smoothing
parameter (rather than b and k) could be formulated, but the rule
as presented here makes it simple for the modeler to set a desired
target activity l, and is in any case relatively insensitive to the values of the smoothing parameters.
2.7. Learning
Initial connection field weights are random within a twodimensional Gaussian envelope. E.g., for a postsynaptic (target)
neuron j located at sheet coordinate (0, 0), the weight xij,p from
presynaptic unit i in projection p is:
xij;p
1
x2 þ y 2
¼
u exp Z xp
2r2p
!
ð7Þ
where (x, y) is the sheet-coordinate location of the presynaptic neuron i, u is a scalar value drawn from a uniform random distribution
for the afferent and lateral inhibitory projections (p = A, I), rp determines the width of the Gaussian in sheet coordinates, and Zx is a
constant normalizing term that ensures that the total of all weights
xij to neuron j in projection p is 1.0. Weights for each projection are
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
6
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
only defined within a specific maximum circular radius rp; they are
considered zero outside that radius.
Each connection weight xij from unit i to unit j is adjusted using
a simple Hebbian learning rule. This rule results in connections
that reflect correlations between the presynaptic activity and the
postsynaptic response. Hebbian connection weight adjustment
for unit j is dependent on the presynaptic activity gi, the postsynaptic response gj, and the Hebbian learning rate ap:
xij;p ðt 1Þ þ ap gj gi
ðxkj;p ðt 1Þ þ ap gj gk Þ
xij;p ðtÞ ¼ P
ð8Þ
k
Unless it is constrained, Hebbian learning will lead to ever-increasing (and thus unstable) values of the weights. The weights are constrained using divisive postsynaptic weight normalization (Eq. (8)),
which is a simple and well understood mechanism. All afferent connection weights from RGC/LGN sheets are normalized together in
the model, which allows V1 neurons to become selective for any
subset of the RGC/LGN inputs. Weights are normalized separately
for each of the other projections, to ensure that Hebbian learning
does not disrupt the balance between feedforward drive, lateral
and feedback excitation, and lateral and feedback inhibition. Subtractive normalization with upper and lower bounds could be used
instead, but it would lead to binary weights (Miller and MacKay,
1994; Miller, 1994), which is not desirable for a firing-rate model
whose connections represent averages over multiple physical connections. More biologically motivated homeostatic mechanisms
for normalization such as multiplicative synaptic scaling (Turrigiano, 1999) or a sliding threshold for plasticity (Bienenstock et al.,
1982) could be implemented instead, but these have not been
tested so far.
3. Results
In each of the subsections below, results are shown from previously reported simulations using partial or simplified versions of
the full unified GCAL model described above. Models were typically run for 10,000 iterations (where an iteration corresponds to
one complete image presentation, e.g. a visual saccade, covering
a simulation time of 1.0), with structured feature maps and receptive fields gradually appearing as more input patterns are presented. By 10,000, the maps and connection fields have come to
a ‘‘dynamic equilibrium’’, where they are stable as long as the input
statistics are stationary (Miikkulainen et al., 2005), and these are
the results that are shown here.
3.1. Feature maps
For each of the topographic feature maps reported for V1 in
imaging experiments, Fig. 3 shows a typical imaging result from
an animal, plotted above a typical final result from a simulation
using the above framework and including that feature value. For
OR, DR, DY, and CR, black indicates neurons not selective for the
indicated feature, and bright colors indicate highly selective neurons. For CR the color in the plot indicates the preferred color;
the others are false-color plots showing the feature value for each
neuron using the color key adjacent to that panel. Maps were measured by collating responses to sine gratings covering ranges of
values of each feature (Miikkulainen et al., 2005), reproducing typical methods of map measurement in animals (Blasdel, 1992a).
Each of these specific simulations was done with an earlier model
named LISSOM (Miikkulainen et al., 2005), which is similar to the
full GCAL model described above but simplified to include only a
single cortical layer, with direct inhibitory long-range lateral connections, and with modeler-determined thresholds rather than
the homeostatic mechanisms described above (Eq. (6)). Each result
is for a model including only a subset of the subcortical sheets
shown in Fig. 1. Specifically, the model (X, Y) and OR map simulations use only a single pair of monochromatic On, Off RGC/LGN
sheets at a fixed size, the OD and DY maps are similar to OR but include an additional pair for the other eye, the DR and TF maps are
similar to OR but include three additional pairs with different delays, the SF map is similar to OR but includes three additional pairs
of On, Off cells with different Difference-Of-Gaussians CFs, and the
CR map is similar to OR but includes five additional sheets of color
opponent cells (BY On, RG On, RG Off, GR On, and GR Off).
As described in the indicated original source for each model, the
model results for (X, Y), OR, OD, DR, and SF have been evaluated
against the available animal data, and capture the main aspects
of the feature value coverage and the spatial organization of the
maps (Miikkulainen et al., 2005; Palmer, 2009). For instance, the
OR and DR maps show iso-feature domains, pinwheel centers, fractures, saddle points, linear zones, and an overall ring-shaped Fourier transform, as in the animal maps (Blasdel, 1992a; Weliky et al.,
1996). The maps simulated together (e.g. OR and OD) also tend to
intersect at right angles, such that high-gradient regions in one
map avoid high-gradient regions in others (Miikkulainen et al.,
2005).
These patterns primarily emerge from geometric constraints
on smoothly mapping the range of values for the indicated feature, within a two-dimensional retinotopic map (Miikkulainen
et al., 2005). They are also affected by the relative amount by
which each feature varies in the input dataset, how often each
feature appears, and other aspects of the input image statistics
(Miikkulainen et al., 2005). For instance, orientation maps
trained on natural image inputs develop a preponderance of
neurons with horizontal and vertical orientation preferences, as
seen in ferret maps and in natural images (Bednar and Miikkulainen, 2004; Coppola et al., 1998).
For DY, the model results (Ramtohul, 2006) have not been compared systematically with the small amount of data (barely visible
in the plot) from the one available experimental report on the
organization for disparity preferences (Kara and Boyd, 2009), because the model predated the experiments by several years. A preliminary analysis suggests that the model organization is
comparable in that neurons for disparity tend to occur in small, local regions sensitive to horizontal disparity, but further evaluation
is necessary. The results for color (CR) are preliminary due to ongoing work on this topic, but do show that color-selective neurons
are found in spatially segregated blobs that include multiple color
preferences, as suggested by the imaging data currently available
(Xiao et al., 2007). Results for TF are also preliminary, again because they predated experimental TF maps and have not yet been
characterized against the experimental results.
Overall, where it has been possible to make comparisons, the
separate models have been shown to reproduce the main features
of the experimental data, using a small set of assumptions. In each
case, the model demonstrates how the experimentally measured
map can emerge from Hebbian learning of corresponding patterns
of subcortical and cortical activity. The models thus illustrate how
the same basic, general-purpose adaptive mechanism will lead to
very different organizations, depending on the geometrical and
statistical properties of that feature.
So far, only a subset of these features have been combined into a
single simulation, such as (X, Y), OR, OD, DR, and TF (Bednar and
Miikkulainen, 2006). Future work will focus on showing how all
or nearly all of these results could emerge simultaneously in the
full unified GCAL model. However, note that only a few of these
maps have ever been measured in the same animal, or even the
same species, and thus it is not yet known whether all such maps
are actually present in any single animal. The unified model can be
used to understand how the maps interact, and to make
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
7
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
1
5
6
10
2
3
5
5
5
8
9
7
11
12
4
5
Fig. 3. Simulated vs. real animal V1 maps. Imaging results for 4 mm 4 mm of V1 of the indicated species and from the corresponding LISSOM models of retinotopy (X, Y),
orientation (OR), ocular dominance (OD), motion direction (DR), spatial frequency (SF), temporal frequency (TF), disparity (DY), and color (CR). See the main text for
description. Reprinted from references indicated: 1Bosking et al. (2002). 2Blasdel (1992a). 3Blasdel (1992b). 4Weliky et al. (1996). 5Miikkulainen et al. (2005). 6Xu et al. (2007).
7
Kara and Boyd (2009). 8Xiao et al. (2007). 9Purushothaman et al. (2009). 10Palmer (2009). 11Ramtohul (2006). 12Ball and Bednar (2009).
predictions for the expected organization for maps not yet measured in a particular species.
3.2. Connection patterns
The feature maps described in the previous subsection are summaries of the properties of a set of neurons embedded in a network.
To understand how these properties come about, it is necessary to
look at the patterns of connectivity that underlie them. Fig. 4 shows
examples of such connections from an (X, Y) and OR GCAL simulation
using a single pair of On, Off LGN/RGC sheets (Antolik, 2010). Note
that this model focuses only on the emergence of orientation preferences, not on other dimensions such as spatial frequency, which
would require additional sheets as shown in Fig. 1. The afferent connections that develop reflect the feature preference of the target
neuron, with elongated Gabor-shaped connection fields that respond to oriented edges in the input. Only a single orientation edge
size is represented, because this simulation includes only a single
RGC/LGN cell RF size. The orientation preference suggested by the
afferent weight pattern strongly correlates with the measured orientation map (Antolik, 2010; not shown).
Because the lateral weights, like the afferent weights, are
modified by Hebbian learning, they reflect the correlation patterns between V1 neurons, which are determined both by the
emerging map patterns and by the input statistics. For this OR
map, retinotopy and OR strongly determine the correlations,
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
8
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
Fig. 4. Self-organized projections to V1 L2/3. Results from a GCAL model orientation map with separate V1 L4 and L2/3 regions allowing the emergence of complex cells;
other dimensions like OD, DR, DY, SF, and CR are not included here. (a and b) Connection fields from the LGN On and Off channels to every 20th neuron in the model L4 show
that the neurons develop OR preferences that cover the full range at each retinotopic location. (c) Long-range excitatory lateral connections to those neurons preferentially
come from neurons with similar OR preferences. Here strong weights are colored with the OR preference of the source neuron. Strong weights occur in clumps (appearing as
small dots here) corresponding to an iso-orientation domain (each approximately 0.2–0.3 mm wide); the fact that most of the dots are similar in color for any given neuron
shows that the connections are orientation specific. (d) Enlarged plot from (c) for a typical OR domain neuron that prefers horizontal patterns and receives connections
primarily from other horizontal-preferring neurons (appearing as blobs of red or nearly red colors). (e) OR pinwheel neurons receive connections from neurons with many
different OR preferences. Reprinted from Antolik (2010).
and thus the lateral connection patterns respect both the retinotopic and the orientation maps. Lateral connection patterns are
thus patchy and orientation specific, as seen in tree shrews
and monkeys (Bosking et al., 1997; Sincich and Blasdel, 2001)
(see Fig. 5). For neurons in orientation domains, long-range connections primarily come from neurons with similar orientation
preference, because those neurons were often coactivated with
this neuron during self-organization using natural images. Interestingly, a prediction is that cells near pinwheel fractures, which
are less selective for orientation in the model, will have a much
broader range of input connections, because their activity is correlated with neurons with a wide, non-specific range of orientation preferences. The degree of orientation selectivity of neurons
near pinwheel centers has been controversial, but current evidence is in line with the model results (Nauhaus et al., 2008).
Connectivity patterns in pinwheels have not yet been investigated experimentally, and so the model results represent predictions for future experiments.
When multiple maps are simulated in the same model, the connection patterns respect all maps at once, because there is only one
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
9
Fig. 5. Lateral connections across maps. LISSOM/GCAL neurons each participate in multiple functional maps, but have only a single set of lateral connections. Connections are
strongest from other neurons with similar properties, respecting each of the maps to the degree to which that map affects correlation between neurons. Maps for a combined
LISSOM OR/OD/DR simulation are shown above, with the black outlines indicating the connections to the central neuron (marked with a small black outline) that remain after
weak connections have been pruned. Model neurons connect to other model neurons with similar orientation preference (a) (as in tree shrew, (d)) but even more strongly
respect the direction map (c). This highly monocular unit also connects strongly to the same eye (b), but the more typical binocular cells have wider connection distributions.
(a–c) Reprinted from Bednar and Miikkulainen (2006); (d) reprinted from Bosking et al. (1997).
set of V1 neurons and one set of lateral connections, each determined by Hebbian learning. Fig. 5 shows an example from a combined (X, Y), OR, OD, DR map simulation, which reproduces the
observed connection dependence on the OR map but predicts that
connections will also respect the DR map (as reported by Roerig
and Kao (1999)), and for highly monocular neurons will respect
the OD map as well. The model strongly predicts that lateral connection patterns will respect all other maps that account for a significant fraction of the response variance of the neurons, because
each of those features will thus affect the correlation between
neurons.
3.3. Orientation and phase tuning
Typical map development models focus on reproducing the
map patterns, and are otherwise highly abstract (for review see
Swindale (1996), Goodhill (2007), and Erwin et al. (1995)). These
models are very useful for understanding the process of map formation in isolation, and to explain the geometric properties of
maps. However, a map pattern measured in a real animal is simply
a summary of one property of a complete system that processes visual information, and so a full explanation of the map pattern requires a demonstration of how the map patterns emerge from a
set of neurons that process visual information in a realistic way.
For such a model, the map patterns can be a way to determine if
the underlying model circuit is operating like those in animals,
which is a very different goal than in abstract models of map patterns in isolation. Once such a model exhibits realistic maps, it can
then be tested to determine whether the feature preferences summarized in the map actually represent realistic responses to a visual feature, such as the orientation and phase of a sine grating
test pattern.
For instance, neurons in a model orientation map can be tested
to see if they are actually selective for the orientation of an input
stimulus, and retain that selectivity as contrast is varied (i.e., have
robust contrast-invariant tuning (Sclar and Freeman, 1982)). Many
developmental models (e.g. those based on the elastic net or using
correlation-based learning) cannot be tested with a specific bitmap
input image, and so cannot directly be extended into a model visual system of the type considered here. The results from others
that do allow bitmap input are not typically compared with single-unit data from animals, again because the papers focus on
the map patterns. Yet without contrast-invariant tuning, the response of a neuron would be ambiguous – a strong response would
could indicate either that the preferred orientation is present, or
else that the input simply has very high contrast.
Fig. 6 shows that GCAL model neurons, thanks to the lateral
inhibition implemented at the RGC/LGN level, do retain their orientation tuning as contrast is varied (Antolik and Bednar, 2011). Earlier models such as LISSOM (Miikkulainen et al., 2005) did not have
this property and were thus valid only for a small range of contrasts. In GCAL, lateral inhibition acts like divisive normalization
of the population response, giving similar patterns and levels of
activity regardless of the contrast, which leads to contrast-invariant tuning (Antolik and Bednar, 2011).
In the animal V1 data, tuning for spatial phase (the specific position of an oriented sine grating) is complicated, with simple cells
highly selective for both orientation and spatial phase, and complex cells selective for orientation but not spatial phase (Hubel
et al., 1962). Moreover, the spatial organization is smooth for orientation (Blasdel, 1992a), such that nearby neurons have similar
orientation preferences, but disordered for phase, with nearby neurons having a variety of phase preferences (Aronov et al., 2003; Liu
et al., 1992; Jin et al., 2011) (though the detailed organization for
spatial phase is not yet clear and consistent between studies).
Disorder in the phase map provides a simple, local way to construct complex cells—simply pool outputs from several local simple cells, each with similar orientation preferences but one of a
variety of phase preferences, to construct an orientation-selective
cell relatively invariant to phase (as originally proposed by Hubel
et al. (1962)). However, it is difficult to develop a random phase
map in simulation, because existing developmental models group
neurons by similarity in responses to a visual pattern, while neurons with different phase preferences will by definition respond
to different visual patterns (Antolik and Bednar, 2011). Previous
models for the development of random phase preferences have relied on unrealistic mechanisms such as squaring negative activation levels (Hyvärinen and Hoyer, 2001; Weber et al., 2001),
which force opposite phases to become correlated (and thus
grouped together by developmental models) but have no clear
physical interpretation.
Instead, the GCAL models (Antolik and Bednar, 2011; Antolik,
2010) show how random phase maps could arise in a set of simple
cells (nominally in layer 4) through a small amount of initial disorder in the mapping from the thalamus, which persists because the
model includes strong long-range lateral connectivity only in the
layer 2/3 complex cells rather than in layer 4 (see Fig. 2). The layer
2/3 cells pool from a local patch of layer 4 cells, thus becoming
unselective for phase, but have strong lateral interactions that lead
to well-organized maps in layer 2/3 (which is primarily what is
measured in most optical imaging experiments). Feedback from
layer 2/3 to layer 4 then causes layer 4 cells to develop an orientation map, without disrupting the random phase map. Fig. 7 plots
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
10
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
Fig. 6. Contrast-invariant tuning. Unlike other developmental models such as LISSOM, GCAL shows contrast-invariant tuning, i.e., similar orientation tuning width at different
contrasts, as found in animals (Sclar and Freeman, 1982). Reprinted from Antolik and Bednar (2011).
the results of this process, showing the resulting maps and modulation ratios for each layer. The model predicts a strong spatial
organization for (weak) phase preferences in layer 2/3, and that
the orientation map in layer 4 (not currently measurable with
imaging techniques due to its depth beneath the cortical surface)
is less well ordered than in layer 2/3. Fig. 6 shows that the complex
cells in layer 2/3 retain orientation tuning and contrast invariance,
as expected.
very large range of possible input patterns suffices to develop
map patterns—orientation maps (but not realistic lateral connection patterns) can develop from random noise inputs, abstract spatially localized patterns, retinal wave model patterns, and other
two-dimensional patterns (Miikkulainen et al., 2005). Thus the
emergence of maps and RFs is a robust phenomenon, but the specific patterns of response properties and neural connectivity directly reflect the input scene statistics.
3.4. Dependence on input statistics
3.5. Surround modulation
Because development in the model is driven by input patterns
(whether externally or intrinsically generated), the results depend
crucially on the specific patterns used. At the extreme, models with
two eyes having identical inputs do not develop ocular dominance
maps, and models trained only on static images do not develop
direction maps (Miikkulainen et al., 2005). The specific map patterns also reflect the input statistics, with more neurons responding to horizontal and vertical contours because of the prevalence of
such contours in natural images (Bednar and Miikkulainen, 2004).
Relationships between the maps also reflect the input statistics—
direction maps dominate the overall organization when input patterns are all moving quickly (Miikkulainen et al., 2005), and ocular
dominance maps interact orthogonally with orientation maps only
for some types of eye-specific input differences (Jegelka et al.,
2006). Finally, the lateral connection patterns depend crucially
on the input statistics, with long-range orientation specific connectivity developing only for input datasets with long-range orientation-specific correlations (Miikkulainen et al., 2005). Even so, a
Given a model with realistically patchy, specific lateral connectivity and realistic single-neuron properties, as outlined above, the
patterns of interaction between neurons can be compared with
neurophysiological evidence for surround modulation—influences
on neural responses from distant patterns in the visual field. These
studies can help validate the underlying model circuit, while helping understand how the visual cortex will respond to complicated
patterns such as natural images.
For instance, as the size of a patch of grating is increased, the response of a V1 neuron typically increases at first, reaches a peak, and
then decreases (Sceniak et al., 1999; Sengpiel et al., 1997; Wang
et al., 2009). Similar patterns can be observed in a GCAL-based model
orientation map with complex cells and separate inhibitory and
excitatory subpopulations (Fig. 8 from Antolik (2010); see connectivity in Fig. 2). Small patterns initially activate neurons weakly,
due to low overlap with the afferent receptive fields of layer 4 cells,
but the response increases with larger patterns. For large enough
patterns, lateral interactions are strong and in most locations net
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
11
Fig. 7. Complex and simple cells. Models with layer 4C and layer 2/3 develop both complex and simple cells (Antolik and Bednar, 2011). The layer 2/3 orientation map in (b) is
a good match to the experimental data from imaging the superficial layers. The matching but slightly disordered OR map in (a), the heavily disordered L4C phase map in (c),
and the weak but highly ordered L2/3 phase map in (d) are all predictions of the model, as none of those have been measured in animals. (e) The layer 4 cells are highly
sensitive to phase (and thus ‘‘simple cells’’), while the layer 2/3 cells respond to a wide (though not entirely flat) range of phases (with response plotted on the vertical axis
and spatial phase on the horizontal). The overall histogram of modulation ratios (ranging from perfectly ‘‘complex’’, i.e., insensitive to phase, to perfectly ‘‘simple’’, responding
to one phase only), is bimodal (f) as in macaque (g). Most model simple cells are in layer 4, and most complex cells are in layer 2/3, but feedback causes the two categories to
overlap. (a–f) Reprinted from Antolik and Bednar (2011); (g) reprinted from Ringach et al. (2002).
Fig. 8. Size tuning. Responses from a particular neuron vary as the input pattern size increases. For a small sine grating (radius 0.2), responses are weak due to low overlap
between the pattern and the neuron’s afferent connection field, even when the position and phase are optimal for that neuron (as here). An intermediate size (0.8) leads to a
peak in response for this neuron, with the maximum ratio between excitation and inhibition. Larger sizes activate a larger area of V1, but responses are sparser due to the
strong lateral inhibition recruited by this salient pattern. For this particular neuron, the response happens to be suppressed for radius 1.8, but note that many other neurons
are still highly active, which is one reason that surround modulation properties are highly variable. The dashed line indicates the 1.0 1.0 area of the retina that is
topographically mapped to the V1 layer 2/3 sheet. Reprinted from Antolik (2010).
inhibitory, causing many neurons to be suppressed (leading to a
subsequent dip in response). These patterns are visualized and
compared with the experimental data in Fig. 9. The model
demonstrates that the lateral interactions are sufficient to account
for typical size tuning effects, and also accounts for less commonly
reported effects that result from neurons with different specific
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
12
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
Fig. 9. Diversity in size tuning. Plots A–F reprinted from Antolik (2010) show how the neural response varies as the sine grating radius increases, for 50% (blue) and 100% (red)
contrasts, for six example neurons. Those on the top row are the most common type (37% of model neurons measured), and are a good match to the phenomena reported in
experimental studies in macaque (G, reprinted from Sceniak et al., 1999) and cat (H, reprinted from Sengpiel et al., 1997, and I, reprinted from Wang et al., 2009), I (Wang
et al., 2009)). Those in the middle row occur less often and are less commonly reported in experimental studies, such as larger responses to low contrasts at high radii, but
examples of each such pattern can be found in the experimental results shown here. The model predicts that these variations in surround tuning properties are real, not just
noise or an experimental artifact, and that they derive from the many possible interactions between a diverse set of neurons in the map. Reprinted from Antolik (2010).
self-organized patterns of lateral connectivity. The model thus accounts both for the typical pattern of size tuning, and explains why
such a diversity of patterns is observed in animals.
The effects of the self-organized lateral connections are even
more evident when orientation-specific surround modulation is
considered. Because the connections respect the orientation map
(Figs. 4 and 5), interactions change depending on the relative orientation of center and surround elements (Fig. 10). Moreover, because these patterns also vary depending on the location of the
neuron in the map, a variety of patterns of interaction are seen, just
as has been reported in experimental studies. Fig. 11 illustrates
some of these relationships, but many more such relationships
are possible—any property of the neurons that varies systematically across the cortical surface will affect the pattern of interactions, as long as it changes the correlation between neurons
during the history of visual experience. These results suggest both
that lateral interactions may underlie many of the observed surround modulation effects, and also that the diversity of observed
effects can at least in part be traced to the diversity of lateral connection patterns, which in turn is a result of the sequences of activations of the neurons during development.
3.6. Aftereffects
The previous sections have focused on the network organization and operation after Hebbian learning can be considered to
be completed. However, the visual system is continually adapting
to the visual input even during normal visual experience, resulting in phenomena such as visual aftereffects (Thompson and Burr,
2009). To investigate whether and how this adaptation differs
from long-term self-organization, we tested LISSOM and GCALbased models with stimuli used in visual aftereffect experiments
(Bednar and Miikkulainen, 2000; Ciroux, 2005). Surprisingly, the
same Hebbian equations that allow neurons and maps to develop
selectivity also lead to realistic aftereffects, such as for orientation
and color (Fig. 12). In the model, we assume that connections
adapt during normal visual experience just as they do in simulated long-term development, albeit with a lower learning rate
appropriate for adult vision. If so, neurons that are coactive during a particular visual stimulus (such as a vertical grating) will
become slightly more strongly laterally connected as they adapt
to that pattern. Subsequently, the response to that pattern will
be reduced, due to increased lateral excitation that leads to net
(disynaptic) lateral inhibition for high contrast patterns like those
in the aftereffect studies. Assuming a population decoding model
such as the vector sum (Bednar and Miikkulainen, 2000), there
will be no change in the perceived orientation of the adaptation
pattern, but the perceived value of a nearby orientation will be
repelled away from the adapting stimulus, because the neurons
activated during adaptation now inhibit each other more strongly,
shifting the population response. These changes are the direct result of Hebbian learning of intracortical connections, as can be
shown by disabling learning for all other connections and observing no change in the overall behavior.
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
13
Fig. 10. Orientation-contrast tuning curves (OCTCs). Surround modulation changes as the orientation of a surrounding annulus is varied relative to a center sine-grating patch
(as in the sample stimulus at the bottom right). In each graph A–F reprinted from Antolik (2010), red is the orientation tuning curve (as in Fig. 6) for the given neuron (with
just the center grating patch), blue is for surround contrast 50%, and green is for surround contrast 100%. Top row: typically (51% of model neurons tested), a collinear
surround is suppressive for these contrasts, but the surround becomes less suppressive as the surround orientation is varied (as for cat (G, reprinted from Sengpiel et al., 1997)
and macaque (H, reprinted from Jones et al., 2002)). Middle row: Other patterns seen in the model include high responses at diagonals (D, 20%, as seen in Sengpiel et al.
(1997)), strongest suppression not collinear (E, as seen in Jones et al. (2002)), and facilitation for all orientations (F, 5%). The relatively rare pattern in F has not been reported
in existing studies, and thus constitutes a prediction. In each case the observed variability is a consequence of the model’s Hebbian learning that leads to a diversity of
patterns of lateral connectivity, rather than noise or experimental artifacts.
Interestingly, for distant orientations, the human data suggests
an attractive effect, with a perceived orientation shifted towards
the adaptation orientation (Mitchell and Muir, 1976). The model
reproduces this feature as well, and provides the novel explanation
that this indirect effect is due to the divisive normalization term in
the Hebbian learning equation (Eq. (8)). Specifically, when the neurons activated during adaptation increase their mutual inhibition,
the normalization term forces this increase to come at the expense
of connections to other neurons not (or only weakly) activated during adaptation. Those neurons are thus disinhibited, and can respond more strongly than before, shifting the response towards
the adaptation stimulus.
Similar patterns occur for the McCollough Effect (McCollough,
1965) (Fig. 12). Here the adaptation stimulus coactivates neurons
selective for orientation, color, or both, and again the lateral
interactions between all these neurons are strengthened. Subsequent stimuli then appear different in both color and orientation,
in patterns similar to the human data. Interestingly, the
McCollough effect can last for months, which suggests that the
modeled changes in lateral connectivity can become
essentially permanent, though the effects of short-term
exposure typically fade in darkness or in subsequent visual
experience.
Overall, the model suggests that the same process of Hebbian
learning could explain both long-term development and shortterm adaptation, unifying phenomena previously considered distinct. Of course, the biophysical mechanisms may indeed be distinct, potentially operating at different time scales and being
largely temporary rather than the permanent changes found early
in development. Even so, the results here suggest that both early
development and adult short-term adaptation may operate using
similar mathematical principles. How mechanisms for long-term
and short-term plasticity may interact, including possible transitions from long-term to short term plasticity during so-called ‘‘critical periods’’, is an important area for future modeling and
experimental studies.
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
14
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
Fig. 11. Surround population effects. Some of the observed variance in surround modulation effects can be explained based on the properties of the measured model neuron
or its neighbors. These measurements are based only on a relatively small number of model neurons, because each data point requires a several-hour-long computational
experiment to choose the optimal stimuli for testing, but some trends are clear in the data so far. For instance, the modulation ratio and suppression index are somewhat
correlated (r = 0.226, p < 0.07) – i.e., simple cells appear to be suppressed a bit more strongly (left). The amount of local homogeneity is a measure of how slowly the map is
changing around a given neuron. Model neurons in homogeneous regions of the map exhibit lower orientation-contrast suppression (r = 0.362, p < 0.01; middle) but greater
overall surround suppression (r = 0.327, p < 0.05). Although preliminary, these predictions can be tested in animal experiments. Moreover, these analyses suggest that much
of the observed variance in surround modulation properties could be due to network and map effects like these, caused by Hebbian learning, with potentially many more
possible interactions for neurons embedded in multiple overlaid functional maps. Reprinted from Antolik (2010).
Fig. 12. Aftereffects as short-term self-organization. (a) While the fully organized network is repeatedly presented patterns with the same orientation, connection strengths
are updated by Hebbian learning (as during development, but at a lower learning rate). The net effect is increased inhibition, which causes the neurons that responded during
adaptation to respond less afterwards. When the overall response is summarized as a ‘‘perceived value’’ using a vector average, the result is systematic shifts in perception,
such that a previously similar orientation will now seem very different in orientation, while more distant orientations will be unchanged or go in the opposite direction. These
patterns are a close match to results from humans (Mitchell and Muir, 1976), suggesting that short-term and long-term adaptation share similar rules. (b) Similar
explanations apply to the McCollough effect (McCollough, 1965), an orientation-contingent color aftereffect; here the model predicts that lateral connections between
orientation and color-selective neurons cause this effect (Ellis, 1977) (and many others in other maps, such as motion aftereffects). (a) Reprinted from Bednar and
Miikkulainen (2000) and replotting data from Mitchell and Muir (1976); (b) reprinted from Ciroux (2005) and replotting data from Ellis (1977).
4. Discussion and future work
The results reviewed above illustrate a general approach to
understanding the large-scale development, organization, and
function of cortical areas. The models show that a relatively small
number of basic and largely uncontroversial assumptions and principles may be sufficient to explain a very wide range of experimental results from the visual cortex. Even very simple neural units,
i.e., firing-rate point neurons, generically connected into topographic maps with initially random or isotropic weights, can form
a wide range of specific feature preferences and maps via unsupervised normalized Hebbian learning of natural images and spontaneous activity patterns. The resulting maps consist of neurons
with realistic visual response properties, with variability due to visual context and recent history that explains significant aspects of
surround modulation and visual aftereffects. The simulator and
example simulations are freely downloadable from topographica.org (see Fig. 13), allowing any interested researcher to build
on this work.
The long-term goal of this project is to understand whether and
how a single, generic cortical circuit and developmental process
can account for the major known information-processing properties of the visual cortex. If so, such a model would be a general-purpose model of cortical computation, with potentially many
applications beyond computational neuroscience. Similar models
have already been used for other cortical regions, such as rodent
barrel cortex (Wilson et al., 2010). Combining the existing models
into a single, runnable visual system is very much a work in progress, but the results so far suggest that doing so will be both feasible and valuable.
Once a full unified model is feasible, an important test will be to
determine if it can replicate both the feature-based analyses described in most of the sections above, and also findings that the visual cortex acts as a unified map of spatiotemporal energy (Basole
et al., 2003). Although the model results are compatible with the
feature-based view, the actual underlying elements of the model
are better described as nonlinear filters, responding to some local
region of the high-dimensional input space. Replicating the results
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
15
Fig. 13. Topographica simulator. This Topographica session shows a user analyzing a simple three-level model with two LGN/RGC sheets and one V1 sheet. (Clockwise from
top) Displays for activity patterns, gradients, Fourier transforms, CFs for one neuron, the overall network, histograms of orientation preference, orientation maps, and
projections (center) are shown. The simulator allows any number of sheets to be defined, interconnected, and analyzed easily, for any simulation defined as a network of
interacting two-dimensional sheets of neurons.
from Basole et al. (2003) in the model would help unify these two
competing explanations for cortical function, showing how the
same underlying system could have responses that are related both
to stimulus energy and to stimulus features.
Building realistic models of this kind depends on having a realistic model of a subcortical pathway. The approach illustrated in
Fig. 1 allows arbitrarily detailed implementation of subcortical
populations, and allows clear specification of model properties.
However, it also results in a large number of subcortical neural
sheets that can be difficult to simulate, and these also introduce
a significant number of new parameters. An alternative approach
is to model the retina as primarily driven by random wiring, an
idea which has recently been verified to a large extent (Field
et al., 2010). In this approach, a regular hexagonal grid of photoreceptors would be set up initially, and then a wide and continuous
range of retinal ganglion cell types would be created by randomly
connecting photoreceptors over local areas of the grid. This process
would require some parameter tuning to ensure that the results
cover the range of properties seen in animals, but could automatically generate realistic RGCs to drive V1 development.
At present, all of the models reviewed contain feedforward and
lateral connections, but no feedback from higher cortical areas to
V1 or from V1 to the LGN, because such feedback has not been
found necessary to replicate the features surveyed. However, note
that nearly all of the physiological data considered was from anesthetized animals not engaged in any visually mediated behaviors.
Under those conditions, it is not surprising that feedback would
have relatively little effect. Corticocortical and corticothalamic
feedback is likely to be crucial to explain how these circuits oper-
ate during natural vision (Sillito et al., 2006; Thiele et al., 2009),
and determining the form and function of this feedback is an
important aspect of developing a general-purpose cortical model.
Because they focus on long-term development, the models discussed here implement time at a relatively coarse level. Most of the
models update the retinal image only a few times each second, and
thus are not suitable for studying the detailed time course of neural
responses. A fully detailed model would need to include spiking,
which makes most of the analyses presented here more difficult
and much more time consuming, but it is possible to simulate even
quite fine time scales at the level of a peri-stimulus-time histogram
(PSTH). I.e., rather than simulating individual spike events, one can
calibrate model neurons against a PSTH from a single neuron, thus
matching its average temporal response over time, or against the
average population response (e.g. measured by voltage-sensitive
dye (VSD) imaging). Recent work shows that it is possible to replicate quite detailed transient-onset LGN and V1 PSTHs in GCAL
using the same neural architecture as in the models discussed here,
simply by adding hysteresis to control how fast neurons can become excited and by simulating at a detailed 0.5-ms time step (Stevens, 2011). Importantly, the transient responses are a natural
consequence of lateral connections in the model, and do not require a more complex model of each neuron. Future work will calibrate the model responses to the full time course of population
activity using VSD imaging, and investigate how self-organization
is affected by using this finer time scale during long-term
development.
Because GCAL models are driven by afferent activity, the type
and properties of the visual input patterns assumed are important
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
16
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
aspects of each model. Ideally, a model of visual system development in primates would be driven by color, stereo, foveated video
streams replicating typical patterns of eye movements, movements of an animal in its environment, and responses to visual
patterns. Collecting data of this sort is difficult, and moreover
cannot capture any causal or contingent relationships between
the current visual input and the current neural organization that
can affect future eye and organism movements that will then
change the visual input. In the long run, to account for more complex aspects of visual system development such as visual object
recognition and optic flow processing, it will be necessary to
implement the models as embodied, situated agents (Pylyshyn,
2000; Weng et al., 2001) embedded in the real world or in realistic 3D virtual environments. Building such robotic or virtual
agents will add significant additional complexity, however, so it
is important first to see how much of the behavior of V1 neurons
can be addressed by the present ‘‘open-loop’’, non-situated
approach.
As discussed throughout, the main focus of this modeling work
has been on replicating experimental data using a small number of
computational primitives and mechanisms, with a goal of providing a concise, concrete, and relatively simple explanation for a
wide and complex range of experimental findings. A complete
explanation of visual cortex development and function would go
even further, demonstrating more clearly why the cortex should
be built in this way, and precisely what information-processing
purpose this circuit performs. For instance, realistic RFs can be obtained from ‘‘normative’’ models embodying the idea that the cortex is developing a set of basis functions to represent input
patterns faithfully, with only a few active neurons (Olshausen
and Field, 1996; Bell and Sejnowski, 1997; Hyvärinen and Hoyer,
2001; Rehn and Sommer, 2007), maps can emerge by minimizing
connection lengths in the cortex (Koulakov and Chklovskii, 2001),
and lateral connections can be modeled as decorrelating the input
patterns (Barlow and Földiák, 1989; Dong, 1995). The GCAL model
can be seen as a concrete, mechanistic implementation of these
ideas, showing how a physically realizable local circuit could develop RFs with good coverage of the input space, via lateral interactions that also implement sparsification via decorrelation
(Miikkulainen et al., 2005). Making more explicit links between
mechanistic models like GCAL and normative theories is an important goal for future work. Meanwhile, there are many aspects of
cortical function not explained by current normative models. The
focus of the current line of research is on first capturing those phenomena in a mechanistic model, so that researchers can then build
deeper explanations for why these computations are useful for the
organism.
As previously emphasized, many of the individual results
found with GCAL can also be obtained using other modeling approaches, which can be complementary to the processes modeled
by GCAL. For instance, it is possible to generate orientation maps
without any activity-dependent plasticity, through the initial wiring pattern between the retina and the cortex (Ringach, 2007;
Paik and Ringach, 2011) or within the cortex itself (Grabska-Barwinska and von der Malsburg, 2008). Such an approach cannot
explain subsequent experience-dependent development, whereas
the Hebbian approach of GCAL can explain both the initial map
and later plasticity, but it is of course possible that the initial
map and plasticity occur via different mechanisms. Other models
are based on abstractions of some of the mechanisms in GCAL (Yu
et al., 2005; Farley et al., 2007; Obermayer et al., 1990; Wolf and
Geisel, 2003), operating similarly but at a higher level. GCAL is
not meant as a competitor to such models, but as a concrete,
physically realizable implementation of those ideas, forming a
prototype of both the biological system and potential future artificial vision systems.
5. Conclusions
The GCAL model results suggest that it will soon be feasible to
build a single model visual system that will account for a very large
fraction of the visual response properties, at the firing rate level, of
V1 neurons in a particular species. Such a model will help researchers make testable predictions to drive future experiments to
understand cortical processing, as well as determine which properties require more complex approaches, such as feedback, attention,
and detailed neural geometry and dynamics. The model suggests
that cortical neurons develop to cover the typical range of variation
in their thalamic inputs, within the context of a smooth, multidimensional topographic map, and that lateral connections store
pairwise correlations and use this information to modulate responses to natural scenes, dynamically adapting to both long-term
and short-term visual input statistics.
Because the model cortex starts without any specialization for
vision, it represents a general model for any cortical region, and
is also an implementation for a generic information processing device that could have important applications outside of neuroscience. By integrating and unifying a wide range of experimental
results, the model should thus help advance our understanding
of cortical processing and biological information processing in
general.
Acknowledgements
Thanks to all of the collaborators whose modelling work is reviewed here, and to the members of the Developmental Computational Neuroscience research group, the Institute for Adaptive and
Neural Computation, and the Doctoral Training Centre in Neuroinformatics, at the University of Edinburgh, for discussions and feedback on many of the models. This work was supported in part by
the UK EPSRC and BBSRC Doctoral Training Centre in Neuroinformatics, under grants EP/F500385/1 and BB/F529254/1, and by the
US NIMH grant R01-MH66991. Computational resources were provided by the Edinburgh Compute and Data Facility (ECDF).
References
Adelson, E.H., Bergen, J.R., 1985. Spatiotemporal energy models for the perception of
motion. Journal of the Optical Society of America A 2, 284–299, <http://
web.mit.edu/persci/people/adelson/pub_pdfs/spatio85.pdf>.
Albrecht, D.G., Geisler, W.S., 1991. Motion selectivity and the contrast-response
function of simple cells in the visual cortex. Visual Neuroscience 7, 531–546,
<http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=1772804>.
Alitto, H.J., Usrey, W.M., 2008. Origin and dynamics of extraclassical suppression in
the lateral geniculate nucleus of the macaque monkey. Neuron 57 (1), 135–146,
<http://dx.doi.org/10.1016/j.neuron.2007.11.019>.
Antolik, J., 2010. Unified developmental model of maps, complex cells and surround
modulation in the primary visual cortex. Ph.D. thesis, School of Informatics, The
University of Edinburgh, Edinburgh, UK. <http://hdl.handle.net/1842/4875>.
Antolik, J., Bednar, J.A., 2011. Development of maps of simple and complex cells in
the primary visual cortex. Frontiers in Computational Neuroscience 5, 17,
<http://dx.doi.org/10.3389/fncom.2011.00017>.
Aronov, D., Reich, D.S., Mechler, F., Victor, J.D., 2003. Neural coding of spatial phase
in V1 of the macaque monkey. Journal of Neurophysiology 89 (6), 3304–3327,
<http://dx.doi.org/10.1152/jn.00826.2002>.
Ball, C.E., Bednar, J.A., 2009. A self-organizing model of color, ocular dominance, and
orientation selectivity in the primary visual cortex. In: Society for Neuroscience
Abstracts, Society for Neuroscience. <http://www.sfn.org> (Program No. 756.9).
Barlow, H.B., Földiák, P., 1989. Adaptation and decorrelation in the cortex. In:
Durbin, R., Miall, C., Mitchison, G. (Eds.), The Computing Neuron. AddisonWesley, Reading, MA, pp. 54–72.
Basole, A., White, L.E., Fitzpatrick, D., 2003. Mapping multiple features in the
population response of visual cortex. Nature 424 (6943), 986–990, <http://
dx.doi.org/10.1038/nature01721>.
Bednar, J.A., Miikkulainen, R., 2000. Tilt aftereffects in a self-organizing model of the
primary visual cortex. Neural Computation 12 (7), 1721–1740, <http://
nn.cs.utexas.edu/keyword?bednar:nc00>.
Bednar, J.A., Miikkulainen, R., 2003. Self-organization of spatiotemporal receptive fields
and laterally connected direction and orientation maps. Neurocomputing 52–54,
473–480, <http://nn.cs.utexas.edu/keyword? bednar :neuro computing03>.
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
Bednar, J.A., Miikkulainen, R., 2004. Prenatal and postnatal development of laterally
connected orientation maps. In: Computational Neuroscience: Trends in
Research, 2004, pp. 985–992. <http://nn.cs.utexas.edu/keyword?bednar:
neurocomputing04-or>.
Bednar, J.A., Miikkulainen, R., 2004. Prenatal and postnatal development of laterally
connected orientation maps. Neurocomputing 58–60, 985–992, <http://
nn.cs.utexas.edu/keyword?bednar:neurocomputing04-or>.
Bednar, J.A., Miikkulainen, R., 2006. Joint maps for orientation, eye, and direction
preference in a self-organizing model of V1. Neurocomputing 69 (10–12),
1272–1276, <http://nn.cs.utexas.edu/keyword?bednar:neurocomputing06>.
Bednar, J.A., Kelkar, A., Miikkulainen, R., 2004. Scaling self-organizing maps to
model large cortical networks. Neuroinformatics 2, 275–302, <http://
nn.cs.utexas.edu/keyword?bednar:neuroinformatics04>.
Bell, A.J., Sejnowski, T.J., 1997. The ‘‘independent components’’ of natural scenes are
edge filters. Vision Research 37, 3327–3338, <http://citeseer.nj.nec.com/
bell97independent.html>.
Bienenstock, E.L., Cooper, L.N., Munro, P.W., 1982. Theory for the development of neuron
selectivity: orientation specificity and binocular interaction in visual cortex. The
Journal of Neuroscience 2, 32–48, <http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_ uids=7054394>.
Binzegger, T., Douglas, R.J., Martin, K.A.C., 2009. Topology and dynamics of the
canonical circuit of cat V1. Neural Networks 22 (8), 1071–1078, <http://
dx.doi.org/10.1016/j.neunet.2009.07.011>.
Blasdel, G.G., 1992a. Orientation selectivity, preference, and continuity in monkey
striate cortex. The Journal of Neuroscience 12, 3139–3161, <http://
www.jneurosci.org/cgi/content/abstract/12/8/3139>.
Bonin, V., Mante, V., Carandini, M., 2005. The suppressive field of neurons in lateral
geniculate nucleus. Journal of Neuroscience 25, 10844–10856, <http://
dx.doi.org/10.1523/JNEUROSCI.3562-05.2005>.
Bosking, W.H., Zhang, Y., Schofield, B.R., Fitzpatrick, D., 1997. Orientation selectivity
and the arrangement of horizontal connections in tree shrew striate cortex. The
Journal of Neuroscience 17 (6), 2112–2127, <http://www.jneurosci.org/cgi/
content/full/17/6/2112>.
Bosking, W.H., Crowley, J.C., Fitzpatrick, D., 2002. Spatial coding of position and
orientation in primary visual cortex. Nature Neuroscience 5 (9), 874–882,
<http://dx.doi.org/10.1038/nn908>.
Chance, F.S., Nelson, S.B., Abbott, L.F., 1999. Complex cells as cortically amplified
simple
cells.
Nature
Neuroscience
2
(3),
277–282,
<http://
www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=10195222>.
Ciroux, J., 2005. Simulating the McCollough effect in a self-organizing model of the
primary visual cortex, Master’s thesis, The University of Edinburgh, Scotland,
UK. <http://www.inf.ed.ac.uk/publications/thesis/online/IM050308.pdf>.
Coppola, D.M., White, L.E., Fitzpatrick, D., Purves, D., 1998. Unequal representation
of cardinal and oblique contours in ferret visual cortex. Proceedings of the
National Academy of Sciences of United States of America 95 (5), 2621–2623,
<http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=9482936>.
Dacey, D.M., 2000. Parallel pathways for spectral coding in primate retina. Annual
Review of Neuroscience 23, 743–775, <http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=10845080>.
De Paula, J.B. 2007. Modeling the self-organization of color selectivity in the visual
cortex, Ph.D. thesis, Department of Computer Sciences, The University of Texas
at Austin, Austin, TX. <http://nn.cs.utexas.edu/keyword?depaula:phd07>
Dong, D.W., 1995. Associative decorrelation dynamics: a theory of self-organization
and optimization in feedback networks. In: Tesauro, G., Touretzky, D.S., Leen,
T.K. (Eds.), Advances in Neural Information Processing Systems, vol. 7. MIT
Press, Cambridge, MA, pp. 925–932, <ftp://ftp.ci.tuwien.ac.at/pub/texmf/bibtex/
nips-7.bib>.
Ellis, S.R., 1977. Orientation selectivity of the McCollough effect: analysis by
equivalent contrast transformation. Perception and Psychophysics 22 (6), 539–
544.
Erwin, E., Obermayer, K., Schulten, K.J., 1995. Models of orientation and ocular
dominance columns in the visual cortex: a critical comparison. Neural
Computation
7
(3),
425–468,
<http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=8935959>.
Farley, B.J., Yu, H., Jin, D.Z., Sur, M., 2007. Alteration of visual input results in a
coordinated reorganization of multiple visual cortex maps. The Journal of
Neuroscience
27
(38),
10299–10310,
<http://dx.doi.org/10.1523/
JNEUROSCI.2257-07.2007>.
Felisberti, F., Derrington, A.M., 1999. Long-range interactions modulate the contrast
gain in the lateral geniculate nucleus of cats. Visual Neuroscience 16, 943–956.
Field, G.D., Gauthier, J.L., Sher, A., Greschner, M., Machado, T.A., Jepson, L.H., Shlens,
J., Gunning, D.E., Mathieson, K., Dabrowski, W., Paninski, L., Litke, A.M.,
Chichilnisky, E.J., 2010. Functional connectivity in the retina at the resolution
of photoreceptors. Nature 467 (7316), 673–677, <http://dx.doi.org/10.1038/
nature09424>.
Gilbert, C.D., Wiesel, T.N., 1989. Columnar specificity of intrinsic horizontal and
corticocortical connections in cat visual cortex. The Journal of Neuroscience 9,
2432–2442, <http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=retrieve&
db=pubmed&dopt=abstract&list_uids=2746337>.
Goodhill, G.J., 2007. Contributions of theoretical modeling to the understanding of
neural map development. Neuron 56 (2), 301–311, <http://dx.doi.org/10.1016/
j.neuron.2007.09.027>.
Grabska-Barwinska, A., von der Malsburg, C., 2008. Establishment of a scaffold for
orientation maps in primary visual cortex of higher mammals. The Journal of
17
Neuroscience 28 (1), 249–257, <http://dx.doi.org/10.1523/JNEUROSCI.551406.2008>.
Hubel, D.H., Wiesel, T.N., 1962. Receptive fields, binocular interaction and
functional architecture in the cat’s visual cortex. The Journal of Physiology
160, 106–154, <http://www.pubmedcentral.nih.gov/articlerender.fcgi? tool=
pubmed&pubmedid=14449617>.
Hyvärinen, A., Hoyer, P.O., 2001. A two-layer sparse coding model learns simple and
complex cell receptive fields and topography from natural images. Vision
Research 41 (18), 2413–2423, <http://www.sciencedirect.com/science/article/
B6T0W-43GCBN2-B/1/94ef0f6b0d8cf1d97998c9e4c392e0c2>.
Jegelka, S., Bednar, J.A., Miikkulainen, R., 2006. Prenatal development of ocular dominance
and orientation maps in a self-organizing model of V1. Neurocomputing 69, 1291–
1296, <http://nn.cs.utexas.edu/keyword? jegelka:cns05>.
Jin, J., Wang, Y., Swadlow, H.A., Alonso, J.M., 2011. Population receptive fields of ON
and OFF thalamic inputs to an orientation column in visual cortex. Nature
Neuroscience 14 (2), 232–238, <http://dx.doi.org/10.1038/nn.2729>.
Jones, H.E., Wang, W., Sillito, A.M., 2002. Spatial organization and magnitude of
orientation contrast interactions in primate V1. Journal of Neurophysiology 88
(5), 2796–2808, <http://dx.doi.org/10.1152/jn.00403.2001>.
Kara, P., Boyd, J.D., 2009. A micro-architecture for binocular disparity and ocular
dominance in visual cortex. Nature 458 (7238), 627–631, <http://dx.doi.org/
10.1038/nature07721>.
Klug, K., Herr, S., Ngo, I.T., Sterling, P., Schein, S., 2003. Macaque retina contains an Scone OFF midget pathway. Journal of Neuroscience 23, 9881–9887.
Koulakov, A.A., Chklovskii, D.B., 2001. Orientation preference patterns in
mammalian visual cortex: a wire length minimization approach. Neuron 29,
519–527. U, <http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=retrieve
&db=pubmed&dopt=abstract&list_uids=11239440>.
Law, J.S., 2009. Modeling the development of organization for orientation
preference in primary visual cortex. Ph.D. thesis, School of Informatics, The
University of Edinburgh, Edinburgh, UK. <http://hdl.handle.net/1842/3935>.
Law, J.S., Antolik, J., Bednar, J.A., 2011. Mechanisms for stable and robust
development of orientation maps and receptive fields. Tech. rep., School of
Informatics, The University of Edinburgh, EDI-INF-RR-1404. <http://
www.inf.ed.ac.uk/publications/report/1404.html>.
Liu, Z., Gaska, J.P., Jacobson, L.D., Pollen, D.A., 1992. Interneuronal interaction
between members of quadrature phase and anti-phase pairs in the cat’s visual
cortex. Vision Research 32 (7), 1193–1198.
McCollough, C., 1965. Color adaptation of edge-detectors in the human visual system.
Science
149
(3688),
1115–1116,
<http://links.jstor.org/sici?sici=00368075%2819650903%293%3A149%3A3688%3C1115%3ACAOEIT%3E2.0.CO%3B2-7>.
Miikkulainen, R., Bednar, J.A., Choe, Y., Sirosh, J., 2005. Computational Maps in the
Visual Cortex. Springer, Berlin, <http://computationalmaps.org>.
Miller, K.D., 1994. A model for the development of simple cell receptive fields and the
ordered arrangement of orientation columns through activity-dependent competition
between ON- and OFF-center inputs. The Journal of Neuroscience 14, 409–441,
<http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd
=retrieve&db=pubmed&dopt=abstract&list_uids=8283248>.
Miller, K.D., MacKay, D.J.C., 1994. The role of constraints in Hebbian learning. Neural
Computation 6, 100–126, <http://wol.ra.phy.cam.ac.uk/mackay/abstracts/
constraints.html>.
Mitchell, D.E., Muir, D.W., 1976. Does the tilt aftereffect occur in the oblique
meridian? Vision Research 16, 609–613, <http://dx.doi.org/10.1016/00426989(76)90007-9>.
Nauhaus, I., Benucci, A., Carandini, M., Ringach, D.L., 2008. Neuronal selectivity and
local map structure in visual cortex. Neuron 57 (5), 673–679, <http://dx.doi.org/
10.1016/j.neuron.2008.01.020>.
Obermayer, K., Ritter, H., Schulten, K.J., 1990. A principle for the formation of the
spatial structure of cortical feature maps. Proceedings of the National Academy
of Sciences of United States of America 87, 8345–8349, <http://www.pnas.org/
cgi/content/abstract/87/21/8345>.
Olmos, A., Kingdom, F.A.A., 2004. McGill calibrated colour image database. <http://
tabby.vision.mcgill.ca>.
Olshausen, B.A., Field, D.J., 1996. Emergence of simple-cell receptive field properties
by learning a sparse code for natural images. Nature 381, 607–609, <http://
www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=8637596>.
Paik, S.-B., Ringach, D.L., 2011. Retinal origin of orientation maps in visual cortex.
Nature Neuroscience 14 (7), 919–925, <http://dx.doi.org/10.1038/nn.2824>.
Palmer, C.M., 2009. Topographic and laminar models for the development and
organisation of spatial frequency and orientation in V1, Ph.D. thesis, School of
Informatics, The University of Edinburgh, Edinburgh, UK. <http://
hdl.handle.net/1842/4114>.
Purushothaman, G., Khaytin, I., Casagrande, V.A., 2009. Quantification of optical images
of cortical responses for inferring functional maps. Journal of Neurophysiology 101
(5), 2708–2724, <http://dx.doi.org/10.1152/jn.90696.2008>.
Pylyshyn, Z.W., 2000. Situating vision in the world. Trends in Cognitive Sciences 4 (5),
197–207, <http://www.biomednet.com/library/abstract/TICS.etd00309_ 13646
613_v0004i05_00001477>.
Ramtohul, T. 2006. A self-organizing model of disparity maps in the primary visual
cortex. Master’s thesis, The University of Edinburgh, Scotland, UK. <http://
www.inf.ed.ac.uk/publications/thesis/online/IM060400.pdf>.
Rehn, M., Sommer, F.T., 2007. A network that uses few active neurones to code
visual input predicts the diverse shapes of cortical receptive fields. Journal of
Computational Neuroscience 22 (2), 135–146, <http://dx.doi.org/10.1007/
s10827-006-0003-9>.
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001
18
J.A. Bednar / Journal of Physiology - Paris xxx (2012) xxx–xxx
Ringach, D.L., 2007. On the origin of the functional architecture of the cortex. PLoS
One 2 (2), e251, <http://dx.doi.org/10.1371/journal.pone.0000251>.
Ringach, D.L., Shapley, R.M., Hawken, M.J., 2002. Orientation selectivity in macaque
V1: diversity and laminar dependence. The Journal of Neuroscience 22 (13),
5639–5651, <http://www.jneurosci.org/content/22/13/5639.long>.
Roerig, B., Kao, J.P., 1999. Organization of intracortical circuits in relation to
direction preference maps in ferret visual cortex. The Journal of Neuroscience
19 (24), RC44, <http://www.jneurosci.org/content/19/24/RC44.long>.
Saul, A.B., Humphrey, A.L., 1992. Evidence of input from lagged cells in the lateral
geniculate nucleus to simple cells in cortical area 17 of the cat. Journal of
Neurophysiology 68 (4), 1190–1208, <http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=1432077>.
Sceniak, M.P., Ringach, D.L., Hawken, M.J., Shapley, R., 1999. Contrast’s effect on
spatial summation by macaque V1 neurons. Nature Neuroscience 2, 733–739.
Sclar, G., Freeman, R.D., 1982. Orientation selectivity in the cat’s striate cortex is
invariant with stimulus contrast. Experimental Brain Research 46, 457–461,
<http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=7095050>.
Sengpiel, F., Sen, A., Blakemore, C., 1997. Characteristics of surround inhibition in
cat area 17. Experimental Brain Research 116 (2), 216–228.
Sillito, A.M., Cudeiro, J., Jones, H.E., 2006. Always returning: feedback and sensory
processing in visual cortex and thalamus. Trends in Neuroscience 29 (6), 307–
316, <http://dx.doi.org/10.1016/j.tins.2006.05.001>.
Sincich, L.C., Blasdel, G.G., 2001. Oriented axon projections in primary visual cortex
of the monkey. The Journal of Neuroscience 21, 4416–4426, <http://
www.jneurosci.org/cgi/content/abstract/21/12/4416>.
Stevens, J.-L., 2011. A temporal model of neural activity and VSD response in the
primary visual cortex. Master’s thesis, The University of Edinburgh, Scotland,
UK. <http://www.inf.ed.ac.uk/publications/thesis/online/IT111096.pdf>.
Stockman, A., Sharpe, L.T., 2000. The spectral sensitivities of the middle- and longwavelength-sensitive cones derived from measurements in observers of known
genotype. Vision Research 40, 1711–1737.
Stockman, A., Sharpe, L.T., Fach, C., 1999. The spectral sensitivity of the human
short-wavelength sensitive cones derived from thresholds and color matches.
Vision Research 39, 2901–2927.
Swindale, N.V., 1996. The development of topography in the visual cortex: a review
of models. Network: Computation in Neural Systems 7, 161–247, <http://
www.iop.org/EJ/S/3/251/kxtGlJZKth350bhDwPkibw/article/0954-898X/7/2/
002/ne62r2.pdf>.
Thiele, A., Pooresmaeili, A., Delicato, L.S., Herrero, J.L., Roelfsema, P.R., 2009. Additive
effects of attention and stimulus contrast in primary visual cortex. Cerebral
Cortex 19 (12), 2970–2981, <http://dx.doi.org/10.1093/cercor/bhp070>.
Thompson, P., Burr, D., 2009. Visual aftereffects. Current Biology 19 (1), R11–14,
<http://dx.doi.org/10.1016/j.cub.2008.10.014>.
Turrigiano, G.G., 1999. Homeostatic plasticity in neuronal networks: the more
things change, the more they stay the same. Trends in Neurosciences 22 (5),
221–227, <http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd =retrieve&db=
pubmed&dopt=abstract&list_uids=10322495>.
Vidyasagar, T.R., Kulikowski, J.J., Lipnicki, D.M., Dreher, B., 2002. Convergence of
parvocellular and magnocellular information channels in the primary visual
cortex of the macaque. European Journal of Neuroscience 16 (5), 945–956,
<http://dx.doi.org/10.1046/j.1460-9568.2002.02137.x>.
Wang, C., Bardy, C., Huang, J.Y., FitzGibbon, T., Dreher, B., 2009. Contrast
dependence of center and surround integration in primary visual cortex of
the cat. Journal of Vision 9 (1), 20.1–1, <http://dx.doi.org/10.1167/9.1.20>.
Weber, C., maps, 2001. Self-organization of orientation maps, lateral connections,
and dynamic receptive fields in the primary visual cortex. In: Proceedings of the
International Conference on Artificial Neural Networks. Lecture Notes in
Computer Science, vol. 2130. Springer, Berlin, pp. 1147–1152, <http://
www.bcs.rochester.edu/cweber/publications/ICANN01.ps>.
Weliky, M., Bosking, W.H., Fitzpatrick, D., 1996. A systematic map of direction
preference in primary visual cortex. Nature 379, 725–728, <http://
www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=8602218>.
Weng, J., McClelland, J.L., Pentland, A., Sporns, O., Stockman, I., Sur, M., Thelen, E., 2001.
Autonomous mental development by robots and animals. Science 291 (5504), 599–
600, <http://www.sciencemag.org/cgi/content/full/291/5504/599>.
Wilson, S.P., Law, J.S., Mitchinson, B., Prescott, T.J., Bednar, J.A., 2010. Modeling the
emergence of whisker direction maps in rat barrel cortex. PLoS One 5 (1), e8778,
<http://dx.doi.org/10.1371/journal.pone.0008778>.
Wolf, F., Geisel, T., 2003. Universality in visual cortical pattern formation. Journal of
Physiology – Paris 97 (2–3), 253–264, <http://dx.doi.org/10.1016/
j.jphysparis.2003.09.018>.
Wolfe, J., Palmer, L.A., 1998. Temporal diversity in the lateral geniculate nucleus of
cat. Visual Neuroscience 15 (4), 653–675, <http://www.journals.cambridge.org/
issue_VisualNeuroscience/Vol15No04>.
Wong, R.O.L., 1999. Retinal waves and visual system development. Annual Review
of
Neuroscience
22,
29–47,
<http://www.ncbi.nlm.nih.gov/entrez/
query.fcgi?cmd=retrieve&db=pubmed&dopt=abstract&list_uids=10202531>.
Xiao, Y., Casti, A., Xiao, J., Kaplan, E., 2007. Hue maps in primate striate cortex.
Neuroimage
35
(2),
771–786,
<http://dx.doi.org/10.1016/
j.neuroimage.2006.11.059>.
Xu, X., Anderson, T.J., Casagrande, V.A., 2007. How do functional maps in primary
visual cortex vary with eccentricity? Journal of Comparative Neurology 501 (5),
741–755, <http://dx.doi.org/10.1002/cne.21277>.
Yu, H., Farley, B.J., Jin, D.Z., Sur, M., 2005. The coordinated mapping of visual space
and response features in visual cortex. Neuron 47 (2), 267–280, <http://
dx.doi.org/10.1016/j.neuron.2005.06.011>.
Please cite this article in press as: Bednar, J.A. Building a mechanistic model of the development and function of the primary visual cortex. J. Physiol. (2012),
http://dx.doi.org/10.1016/j.jphysparis.2011.12.001