Download Models of Networks of Neurons Networks of neurons What`s a

Networks of neurons
• Neurons are organized in large networks. A typical neuron is cortex
receives thousands of inputs.
• Aim of modeling networks: explore the computational potential of such
connectivity.
Models of Networks of Neurons
- What computations? (e.g. gain modulation, integration, selective
amplification of some signal, memory etc..)
- What dynamics ? (e.g. spontaneous acticity, variability, oscillations)
•Tools:
- models of neurons and synapses : spiking neurons (IAF) or firing rate
- analytical solutions, numerical integration
1.1 Introduction
3
B
A
dendrite
What’s a network ?
apical
dendrite
~80% excitatory cells (pyramidal neurons),
• In cortex,
1.1 Introduction
3 ~20% inhibitory
soma
neurons (smooth stellate + large variety of other types)/ a.k.a interneurons.
soma
A
B
axon
C
basal
dendrite
dendrite
apical
dendrite
dendrite
axon
soma
axon
collaterals
soma
axon
basal
dendrite
axon
soma
C
dendrite
axon
Figure 1.1:axon
Diagrams of three neurons. A) A cortical pyramidal cell. These are
collaterals
the primary
excitatory neurons of the cerebral cortex. Pyramidal cell axons branch
soma
locally, sending axon collaterals to synapse with nearby neurons, and also project
more distally to conduct
signals
to
other
parts
of
the
brain and nervous system.
axon
B) A Purkinje cell of the cerebellum. Purkinje cell axons transmit the output of
the cerebellar cortex. C) A stellate cell of the cerebral cortex. Stellate cells are
one of a large class of cells that provide inhibitory input to the neurons of the
Figure 1.1: Diagrams of three neurons. A) A cortical pyramidal cell. These are
cerebral
cortex. To give an idea of scale, these figures are magnified about 150 fold.
the primary excitatory neurons of the cerebral cortex. Pyramidal cell axons branch
(Drawings
from Cajal,
1911; figure
fromwith
Dowling,
1992.) and also project
locally, sending
axon collaterals
to synapse
nearby neurons,
more distally to conduct signals to other parts of the brain and nervous system.
B) A Purkinje cell of the cerebellum. Purkinje cell axons transmit the output of
the cerebellar cortex. C) A stellate cell of the cerebral cortex. Stellate cells are
one of a large class of cells that provide inhibitory input to the neurons of the
What’s a network ?
• Laminar Organization.
Cortex is divided into 6 layers.
Models usually pool all layers together.
What’s a network ?
Connectivity
• 3 types of connections: feed-forward, recurrent (lateral), feedback.
• Columnar Organization.
se
on
esp
nR
tio
Neurons in small (30-100 micrometers) columns
perpendicular to the layers
ula
op
B!
P
(across all layers) respond to similar stimulus features.
50
ves
A!
g
nin
Tu
180
r
Cu
90
25
0
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0
! test
V4
0
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s
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90
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Re
nse
spo
IT
0 180
!
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V2
0
! test
V1
Retina
6
FEEDBACK
• Excitatory
• Modulatory influence
V1large
• Convey information from
area of visual field
• V2, MT
• role unclear
V1
LGN
• Excitatory
• precise
topographically
• debated how strong
7
LGN
8
Local
Projections
Long-range Horizontal
Projections
V1
V1
• Excitatory + Inhibitory : lots !!
• < 1 mm
• connectivity depend on distance,
not preference.
• Excitatory - Modulatory
• 6-8 mm (cat, monkey) - Non overlapping RF
• specific/ preferences
LGN
V1: Neurons. !"#$ %&$ '()#*$ +,-.(/-0$ &112$ #3+/.(.,*)$ -#4*,-0$ (-5$ 676$ /-"/8/.,*)$ -#4*,-09$
LGN
9
10
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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 6, NOVEMBER 2003
Network modeling strategies (2)
we choose the ratio of excitatory to inhibitory ne
we make inhibitory synaptic connections stronge
input, each neuron receives a noisy thalamic inp
In principle, one can use RS cells to model
and FS cells to model all inhibitory neurons. Th
heterogeneity (so that different neurons have d
to assign each excitatory cell
spike raster
, where is a random v
tributed on the interval [0,1], and is the neuron
corresponds to regular spiking (RS) cell, and
the chattering (CH) cell. We use
to bias the d
cells. Similarly, each inhibitory cell has
and
.
The model belongs to the class of pulse-cou
(PCNN): The synaptic connection weights bet
given by the matrix
, so that firing
stantaneously changes variable by .
e.g. integrate and fire neurons
• method 1: spiking neurons,
$
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$% D& F
!#D&FA$
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0
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$
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#3= , %
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+#''0$
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, #
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• up to 10,000 neurons.
</G#-$8)$(-$('="(?@4-+./,-A$
0=/>#$."*#0",'5$D?77$;%FA$($0=/>#$/0$#;/..#5A$."#$0=/>#$."*#0",'5$/0$#'#G(.#5$;/;/+>/-<$($*#'(./G#$
U
neuron trace
*#@*(+.,*)$=#*/,5$D0##$C,;#*0$#.$('9$D&EE7F$@,*$5#.(/'0FA$(-5$($P
with electrophysiology, a system where all $;#5/(.#5$(@.#*?")=#*=,'(*/V(./,-$
• advantage: comparison
$
DOLQF$ +,-54+.(-+#$
B(0$ (+./G(.#59$ !"#$ OLQ$ +,-54+.(-+#A$ (OLQD.FA$ ,8#)0$ ."#$ 0(;#$ #W4(./,-$ (0$
neurons can be !recorded"
at all times.
( OLQ (-5$."#$04;$/0$,G#*$."#$/-5#3$!$D."#$+#''X0$,B-$0=/>#0F$*(."#*$
D&SF$#3+#=.$."(.$."#$=*#@(+.,*$/0$
( & % & # ) long
'( + )
simulations,
• difficulties: lots of parameters/assumptions,
$
$
$
$
D&SF$
(!# ! & " $ (!# & *.& % & *# +/ ,
-- #3= ,, %
-- 9 $
,
analysis difficult.
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Fig. 3. Simulation of a network of 1000 randomly coupled spiking neurons.
Top: spike raster shows episodes of alpha and gamma band rhythms (vertical
lines). Bottom: typical spiking activity of an excitatory neuron. All spikes were
first to 30 mV and then to .
equalized at 30 mV by resetting
% Created by Eugene M.
% Excitatory neurons
Ne=800;
re=rand(Ne,1);
[Izhikevitch, 2003]
a=[0.02*ones(Ne,1);
b=[0.2*ones(Ne,1);
c=[-65+15*re.^2;
d=[8-6*re.^2;
S=[0.5*rand(Ne+Ni,Ne),
Izhikevich, F
Inhibitory ne
Ni=200;
ri=rand(Ni,1)
0.02+0.08*ri]
0.25-0.05*ri]
-65*ones(Ni,1
2*ones(Ni,1)]
-rand(Ne+Ni,N
Network modeling strategies (3)
• method 2: reduce the description to describe only rate of spiking r(t),
instead of Vm(t).
τr
dri (t)
= −ri (t) + input(t)
dt
Firing rate model (1)
• each neuron is described at time t by a firing rate v(t).
j=N
!
dvi (t)
wij uj )
τr
= −vi (t) + F (
dt
j=1
• Interpretation: average over time, average over equivalent neurons
• In absence of input, the firing rate relaxes to 0 with a time constant tr - which
also determines how quickly the neuron responds to input.
• The input from a presynaptic neuron is proportional to its firing rate u
• The weight wij determines the strength of connection of neuron j to neuron i
• The total input current is the sum of the input from all external sources.
Firing rate model (2)
• each neuron is described at time t by a firing rate v(t).
τr
j=N
!
dvi (t)
wij uj ) = −vi (t) + F (w.u)
= −vi (t) + F (
dt
dot-product
j=1
• F determines the steady state r as a function of input
• F is called the activation function
• F can be taken as a saturating function, e.g. sigmoid
• F is often chosen to be threshold linear
Network Architectures
• A: Feedforward
τr
N
!
dvi (t)
= −vi (t) + F (
Wij uj (t))
dt
j=1
• B: Recurrent
τr
N
N
!
!
dvi (t)
= −vi (t) + F (
Wij uj (t) +
Mik vk (t))
dt
j=1
k=1
rmax
F (I) =
1 + exp(g(I1/2 − I)
F (I) = G[I − I0 ]+
determined by the equation
τr
dv
= −v + F(h + M · v ) .
dt
(7.11)
Excitatory
- Inhibitory
Network
Neurons are typically
classified
as either
excitatory or inhibitory, meaning
that they have either excitatory or inhibitory effects on all of their postsynaptic
targets.
property
is formalized
Dale’s
law, which
models
haveThis
a single
population
of neuronsinand
the weights
are states that
• Some
a
neuron
cannot
excite
some
of
its
postsynaptic
targets
and
inhibit others.
allowed to be positive and negative.
In terms of the elements of M, this means that for each presynaptic neuron
models represent the excitatory and inhibitory population separately.
• Other
a" , Maa" must have the same sign for all postsynaptic neurons a. To im(more
!biological"
+ richer dynamics).
pose this restriction,
it is convenient to describe excitatory and inhibitory
matrices,
MEE, M
• 4 weight
IE, M
II, MEI
neurons separately.
The
firing-rate
vectors vE and vI for the excitatory and
inhibitory neurons are then described by a coupled set of equations identical in form to equation 7.11,
τE
dvE
= −vE + FE (hE + MEE · vE + MEI · vI )
dt
(7.12)
dvI
= −vI + FI (hI + MIE · vE + MII · vI ) .
dt
(7.13)
Dale’s law
Example:
Orientation selectivity as a model problem:
spiking networks and ring model
excitatoryinhibitory
network
and
τI
There are now four synaptic weight matrices describing the four possible
types of neuronal interactions. The elements of MEE and MIE are greater
than or equal to zero, and those of MEI and MII are less than or equal to
zero. These equations allow the excitatory and inhibitory neurons to have
different time constants, activation functions, and feedforward inputs.
In this chapter, we consider several recurrent network models described
by equation 7.11 with a symmetric weight matrix, Maa" = Ma" a for all a and
neurons
selective
to orientation,
V1’s are: analysis, but symmetric coupling
a" . LGN
Requiring
M toare
be not
symmetric
simplifies
the mathematical
Origin
Orientation
selectivitythat
? neuron a, which is exciit violates Dale’s
law.ofSuppose,
for example,
tatory, and neuron a" , which is inhibitory, are mutually connected. Then,
Draft: December 19, 2000
Theoretical Neuroscience
Model of Hubel and Wiesel (1962)
Text
• Hubel and Wiesel (1962) proposed that the oriented fields of V1 neurons could
be generated by summing the input from appropriately selected LGN neurons.
• The model accounts for selectivity in V1 on the basis of a purely feedforward
architecture.
Ferster and Miller — July 30, 2000
Text
4
Text
V1
Sclar and Freeman, 1982
V1
• Example of a computation, emergence of a new property.
LGN
Figure 1:
neurons
V1 simple cell
A. A map of the receptive field of a simple cell in the cat visual cortex. A light flashed in the ON
subregion (x) or turned off in an OFF region (triangles) excites the cell, while a light flashed in an
OFF region or turned off in the ON region inhibit the cell. Other arrangements of the subregions
are possible, such as a central OFF region and flanking ON regions, or one ON and one OFF
region. B. Hubel and Weisel’s model for how the receptive field of the simple cell can be built
from excitatory input from geniculate relay cells. The simple cell (below right) receives input from
output
output
Connectivity achieves
contrast invariance through
feedforward inhibition
Connectivity sharpens
the tuning curve
input
Due to the precise
organization of the thalamic
Due to input
the imprecise
afferents,
to V1 is
organization
of the
sharply tuned
thalamic afferents, input
to V1 is broadly tuned
output
input
V1
Feedforward
Recurrent
Hubel and Wiesel, 1962;
Troyer, Krukowski,
Priebe and Miller, 1998
Somers, Nelson and Sur 1995;
Sompolinsky and Shapley, 1997
~ 100, 000 synapses
`Feedforward’ model
vs `Recurrent’
~ 1250 conductance
based IAF neurons
Wee, Wie, Wei,
Poisson spike trains
Retina/
LGN
Seriès, Latham and
Pouget - Nature
Neuroscience 2004
ON
OFF
• Explore physiological and anatomical plausibility:
- cortical connectivity scheme,
- thalamocortical connectivity,
- properties of inhibition in Cx (inactivation)
…
(Sompolinsky and Shapley, 1997; Ferster and Miller, 2000).
• Coding efficiency
(are these models making different predictions in terms of
information transmission?)
23
22
7.4 Recurrent Networks
25
1.9. This value, being Network
larger thanmodels
one, would
lead to an unstable network
- summary
in the linear case. While nonlinear networks can also be unstable, the restriction to eigenvalues less than one is no longer the relevant condition.
The Ring Model (1)
In a nonlinear network, the Fourier analysis of the input and output remodels:astoinformative
understand the
of connectivity
• Network
sponses
is no longer
as implications
it is for a linear
network.inDue to
terms
of
computation
and
dynamics.
the rectification, the ν = 0, 1, and 2 Fourier components are all amplified
(figure 7.9D) compared to their input values (figure 7.9C). Nevertheless,
except
rectification,
the nonlinear
network amplifies the inMain
strategies: Spiking
vs Firingrecurrent
rate models.
• 2for
put signal selectively in a similar manner as the linear network.
• The issue of the emergence of orientation selectivity as a model
A Recurrent
of Simple
in Primary
Visual Cortex
problem,Model
extensively
studiedCells
theoretically
and experimentally.
- Two main models: feed-forward and recurrent.
In chapter
2, we
discussed
feedforward
model in
which
- Detailed
spiking
modelsahave
been constructed
which
canthe
be elongated
directly
receptive fields of simple cells in primary visual cortex were formed by
compared to electrophysiology
summing the inputs from lateral geniculate (LGN) neurons with their re- The
same
probleminisalternating
also investigated
with
a firing
rate cells.
model,While
a.k.a.this
ceptive
fields
arranged
rows of
ON
and OFF
thequite
!ringsuccessfully
model".
model
accounts for a number of features of simple cells,
such as orientation tuning, it is difficult to reconcile with the anatomy and
circuitry of the cerebral cortex. By far the majority of the synapses onto
any cortical neuron arise from other cortical neurons, not from thalamic
afferents. Therefore, feedforward models account for the response properties of cortical neurons while ignoring the inputs that are numerically
most prominent. The large number of intracortical connections suggests,
instead, that recurrent circuitry might play an important role in shaping
the responses of neurons in primary visual cortex.
The Ring Model (2)
+
(7.36)
The Ring Model (3)
"#!
• h is input, can be tuned (Hubel Wiesel
h(θ) = c[1 − " + " ∗ cos(2θ)]
/
"$
"#'
scenario) or very broadly tuned.
01!2
Ben-Yishai, Bar-Or, and Sompolinsky (1995) developed a model at the
N neurons,
withfor
preferred
angle, θi ,evenly
distributed
•other
extreme,
which recurrent
connections
are the primary determiners
of
orientation
tuning.
The
model
is
similar
in structure to the model
between −π/2 and π/2
of
equations
7.35
and
7.33,
except
that
it
includes
a global inhibitory inter• Neurons receive thalamic inputs h.
action. In addition, because orientation angles are defined over the range
+ recurrent connections, with excitatory weights between
from −π/2 to π/2, rather than over the full 2π range, the cosine functions
nearby
cells and
inhibitory
cells that
arebasic equation of the
in the model
have
extra weights
factors between
of 2 in them.
The
further
(mexican-hat
model,apart
as we
implement profile)
it, is
!
%
" π/2 "
$
dv(θ)
dθ #
"
"
τr
= −v(θ) + h (θ) +
−λ0 + λ1 cos(2(θ − θ )) v(θ )
dt
−π/2 π
"%
"#&
"#%
"#$
"/
!!"
"
!"
()*+,-.-*(,/!
• The steady-state can be solved analytically.
Model analyzed like a physical system.
!"
-('./*
where v(θ)
is the firing rate of a neuron with preferred orientation θ.
"
!!"
The input to the model represents the orientation-tuned feedforward in!$""
put arising
from ON-center and OFF-center LGN cells responding to an
!$!"
oriented
image. As a function of preferred orientation, the input for an
image!#""
with !!"
orientation
"
!"angle & = 0 is
%&'()*+*'%),!
h (θ) = Ac (1 − ' + ' cos(2θ))
Draft: December 19, 2000
(7.37)
Theoretical Neuroscience
• Model achieves i) orientation selectivity; ii) contrast invariance of tuning, even
if input is very broad.
• The width of orientation selectivity depends on the shape of the mexican-hat,
but is independent of the width of the input.
• Symmetry breaking /Attractor dynamics.
The Ring Model (4)
Attractor Networks
7.4 Recurrent Networks
•Attractor network : a network of neurons, usually recurrently connected, whose time
27
dynamics settle to a stable pattern.
A
B
C
80
firing rate (Hz)
30
v (Hz)
h (Hz)
60
20
10
40
20
• That pattern may be stationary (fixed points), time-varying (e.g. cyclic), or even
80
80%
40%
20%
10%
60
stochastic-looking (e.g., chaotic).
• The particular pattern a network settles to is called its !attractor".
40
•The ring model is called a line (or ring) attractor network. Its stable states are also
20
sometimes referred to as !bump attractors".
0
-40
-20
0
20
θ (deg)
40
0
-40
-20
0
20
θ (deg)
40
0
180
200
220 240
Θ (deg)
Figure 7.10: The effect of contrast on orientation tuning. A) The feedforward input as a function of preferred orientation. The four curves, from top to bottom,
correspond to contrasts of 80%, 40%, 20%, and 10%. B) The output firing rates
in response to different levels of contrast as a function of orientation preference.
These are also the response tuning curves of a single neuron with preferred orientation zero. As in A, the four curves, from top to bottom, correspond to contrasts
of 80%, 40%, 20%, and 10%. The recurrent model had λ0 = 7.3, λ1 = 11, A = 40
Hz, and " = 0.1. C) Tuning curves measure experimentally at four contrast levels
as indicated in the legend. (C adapted from Sompolinsky and Shapley, 1997; based
on data from Sclar and Freeman, 1982.)
Point Attractor
Line Attractor
A Recurrent Model of Complex Cells in Primary Visual Cortex
In the model of orientation tuning discussed in the previous section, recurrent amplification enhances selectivity. If the pattern of network connectivity amplifies nonselective rather than selective responses, recurrent interactions can also decrease selectivity. Recall from chapter 2 that neurons
in the primary visual cortex are classified as simple or complex depending on their sensitivity to the spatial phase of a grating stimulus. Simple
cells respond maximally when the spatial positioning of the light and dark
regions of a grating matches the locations of the ON and OFF regions of
their receptive fields. Complex cells do not have distinct ON and OFF regions in their receptive fields and respond to gratings of the appropriate
orientation and spatial frequency relatively independently of where their
light and dark stripes fall. In other words, complex cells are insensitive to
spatial phase.
Chance, Nelson, and Abbott (1999) showed that complex cell responses
could be generated from simple cell responses by a recurrent network. As
in chapter 2, we label spatial phase preferences by the angle φ. The feedforward input h (φ) in the model is set equal to the rectified response of
a simple cell with preferred spatial phase φ (figure 7.11A). Each neuron
in the network is labeled by the spatial phase preference of its feedforward input. The network neurons also receive recurrent input given by
the weight function M (φ − φ" ) = λ1 /(2πρφ ) that is the same for all conDraft: December 19, 2000
• Model was tested with stimuli containing more
than 1 orientation (crosses)
•Model fails to distinguish angles separated by 30
Theoretical Neuroscience
deg, overestimates larger angles
•spurious attractors with noise
31
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The Ring Model (5): Sustained Activity
Network models - summary
• If recurrent connections are strong enough, the pattern of population
activity once established can become independent of the structure of the
input. It can persists when input is removed.
• A model of working memory ?
• Network models: to understand the implications of connectivity in
terms of computation and dynamics.
• 2 Main strategies: Spiking vs Firing rate models.
• The issue of the emergence of orientation selectivity as a model
problem, extensively studied theoretically and experimentally.
- Two main models: feed-forward and recurrent.
- Detailed spiking models have been constructed which can be directly
compared to electrophysiology
- The same problem is also investigated with a firing rate model, a.k.a.
the !ring model".