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Searching for Principles of Brain Computation
Author: Wolfgang Maass
Affiliation: Graz University of Technology, Institute for Theoretical Computer Science,
Inffeldgasse 16b/I, A-8010 Graz, Austria; e-mail:
Computational role of transient network states
Simultaneous recordings from many neurons constrain computational models
Stochastic state transitions point to Markov chain computations
Ongoing network rewiring and compensation through synaptic sampling
Experimental methods in neuroscience, such as Ca-imaging and recordings with multielectrode arrays, are advancing at a rapid pace. They produce insight into the activity of
large numbers of neurons and plastity processes in the brains of awake and behaving
animals. These data provide new constraints for modelling neural computations and learning
processes that underlie perception, cognition, and behaviour. I will discuss in this short
review four such constraints: Inherent recurrent network activity and heterogeneous dynamic
properties of neurons and synapses, stereotypical spatio-temporal activity patterns, high trialto-trial variability of network responses, and functional stability in spite of permanently
ongoing changes in the network. I will explain that these constraints provide hints to
underlying principles of brain computation and learning.
Constraint/Principle 1: Neural circuits are highly recurrent networks of neurons and
synapses with diverse dynamic properties
Many concepts and computational models in computational neuroscience are strongly
influenced by paradigms for the organization of computation in our current generation of
digital computers. There one typically finds feedforward networks of computational modules
that each carry out a precisely specified subcomputation, and transmit their result to the next
processing stage. Such networks are silent in the absence of network inputs.
In contrast, experimental data suggest that evolution has discovered a completely different
way of organizing computations, that takes place in recurrent rather than feedforward
networks of neurons. From the evolutionary oldest brains, such as brains of hydra (Dupre
and Yuste, 2015), C-elegans (Kato et al, 2015), and zebrafish larvae (Portugues et al.,
2014), to brains of humans (Raichle, 2015) one finds a rich repertoire of temporally extended
activity patterns, both spontaneously and stimulus-evoked, that are characteristic for
recurrent networks. Therefore recent reviews in systems neuroscience (Singer, 2013),
(Yuste, 2015) emphasize the need to understand how highly recurrent networks of neurons
in the brain compute.
The theory of dynamical systems provides interesting concepts, such as criticality, attractors,
and state-trajectories, that may help for that. But the traditional focus of this theory is on
deterministic low-dimensional dynamical systems that consist of simple and homogeneous
dynamic components. In contrast, recurrent neural networks in the brain are stochastic, very
high-dimensional, and consist of different types of neurons and synapses with diverse
dynamical features (Gupta et al., 2000), (Harris and Shepherd, 2015). These experimental
have the unpleasant consequence that one would not even be able to implement in realistic
models for neural circuits standard computing paradigms from computer science and articial
neural networks, since these usually require that the network units are homogeneous and
have no inherent temporal dynamics. Hence a new theory of computation in stochastic highdimensional dynamical systems with heterogeneous dynamic components is needed for
elucidating brain computations.
The liquid computing paradigm (Maass et al., 2002), (Maass et al, 2004), (Legenstein et al.,
2007), (Buanomano et al, 2009) can be viewed as a first step in that direction. Theorem 1 in
(Maass et al., 2002) points to a benefit of diverse dynamic components for computations that
require a fading memory. This computing paradigm also exploits specific computational
benefits of high-dimensional networks:The expressive capability of a linear readout, for
example a projection neuron, is boosted if network inputs are projected nonlinearly into a
high-dimensional state space, see the notion of a kernel in Support Vector Machines
(Bishop, 2006). Remarkably, if one views a recurrent network of neurons as such kernel-like
generic computational preprocessing stage, it is not relevant which nonlinear operations are
carried out by them. It is only required that saliently different network inputs are mapped onto
linearly independent network states. According to this computational model a recurrent
network of neurons carries out two types of generic preprocessing steps for readouts: fading
memory and generic nonlinear preprocessing (Fig. 1a). Obviously this model is very different
from traditional feedforward processing schemes that I had sketched at the beginning of this
section. It also differs from traditional artificial neural network approaches. But a related
computing paradigm for artificial neural networks, that requires --in contrast to the liquid
computing model-- that network units are homogeneous and noise-free, but also employs
random connectivity and has a focus on readouts rather than internal network computations,
had been developed independently in (Jäger, 2001).
An obvious benefit of such processing strategy is that a single recurrent network can
accumulate and preprocess simultaneously a large amount of incoming information, from
which different projection neurons learn to select those features that are relevant for their
specific projection target, see Fig. 1b for a simple demo and (Chen et al., 2013) for related
experimental data.
Mathematical results (Maass et al., 2007) imply that the capability of this computational
model can be substantially enhanced through readout neurons --each trained for specific
tasks-- which project their output through axon collaterals also back into the recurrent
network. In fact, if there is no noise in the network, and readouts can compute arbitrary
continuous functions, then a recurrent network can acquires in this way the computational
power of a unversal Turing machine. Such extreme computational capabilities are not
realistic in the presence of noise, but computer simulations demonstrate that readouts with
feedback can definitely enhance the computational capability of the model: The model can
learn to maintain internal network states (see Fig. 1 c, d), and switch these states as well as
network computations in response to external cues (Fig. 1e).
Figure 1: a) Computational model consistent with constraint 1; b) Demonstration that a generic spike-based
network model with 135 neurons can support diverse linear readouts hat are trained to extract different pieces of
information about two time-varying firing rates f1( t) and f2( t) represented by spiking activity of first 2 and last 2
input neurons shown at the top (see (Maass and Markram, 2007) for details). This provides a new paradigm for
multiplexing of network computations. c) - e): Demonstration of nonfading memory and context-depending
switching of computation on 4 input streams with timevarying Poisson firing rates r1( t),...,r4(t ) shown in c). One
readout with feedback was trained to represent which of r1( t), r2( t) had last exhibited a burst (d; times when r1( t)
had the most recent burst are marked in blue), and another readout was trained to switch between r3( t) + r4( t)
and |r3( t) - r4( t)| in dependence of internal state represented in d).
Readouts with feedback tend to be quite parameter sensitive and require well-tuned learning
methods. Two methods for supervised training of readouts with feedback have been
proposed (Maass et al., 2007) and (Sussillo and Abbott, 2009) build on preceding related
work for artificial neural networks (Jäger and Haas, 2004). These learning methods work well
in simulations, but do not aspire to be biologically realistic A biologically more realistic
learning method based on reinforcement learning has more recently been proposed by
(Hoerzer et al., 2014).
A number of experimental studies have confirmed predictions of the liquid computing model,
such as an inherent fading memory and generic nonlinear preprocessing (Nikolic et al.,
2009), (Klampfl et al., 2012), (Stokes, 2015), generic projection into high dimensional
network states (Rigotti et al., 2013) and state-dependent switching of computational
operations (Mante et al., 2013).
Constraint/Principle 2: Activity in neural network is dominated by variations of a few
stereotypical patterns
Many computational models are based on the assumption that neurons tend to encode
specific features of sensory stimuli or internal variables, likea filter bank. This neuron-centric
view is contradicted by data from simultaneous recordings from large numbers of neurons,
which suggest that their joint activity, both spontaneously and stimulus-evoked, typically
consists of variations of a rather small repertoire of spato-temporal firing patterns that
engage assemblies of hundreds or more neurons. This has been shown for the primary
sensory areas A1 (Luczak et al, 2009), (Bathellier et al., 2012) and V1 (Sadowski and
Maclean, 2014), (Miller et al., 2014). It also has been shown as a result of learning for PFC
(Fujisawa et al., 2008), and PPC (Harvey et al., 2012) . This feature of generic network
activity obviously constrains models for neural computation and coding. Compelling
theoretical models for the computational role of such cell assemblies are still missing, but
many interesting ideas have been proposed (Buzsaki et al., 2010) .
It is not even trivial to produce a neural network model whose response to external inputs is
similarly dominated by variations of a small number of stereotypical patterns. One way of
achieving that is to engage STDP for synaptic connections between pyramidal cells in a
network model that induces competition for firing (and hence for STDP) via strong lateral
inhibition, see (Klampfl and Maass, 2013) for a network model with simplified lateral
inhibition, and Fig. 2 for a newer model where lateral inhibition is carried out by data-based
interactions with inhibitiory neurons (Pokorny et al, 2016).
Viewed as a computational principle, this phenomen facilitates the learning for readout
neurons. Whereas without the occurrence of clear activity patterns for different stimuli a
linear readout has to be trained by supervised learning or reinforcement learning to
recognize and classify incoming stimuli to the network, this task can be carried out readout
neurons that can learn to recognize without supervision or rewards which of a small
repertoire of internal activity patterns is currently on (fig. 2, c,d,e).
A closer look at the experimental data (and the simulation results of Fig. 2 a, b shows that
these stereotypical activity patterns tend to have a clear temporal evolution. Hence they are
reminiscent of the „phase sequences“ proposed by (Hebb, 1949) as a means to integrate
temporally or spatially distributed features into a concept.
Figure 2: a) Emergence of stereotypical assembly responses to repeating external input patterns (blue and green
spike patterns, superimposed by noise spikes shown in black), embedded into Poisson spike input stream with
the same rate. After about100 occurrences of each pattern, stereotypical network responses (assembly
sequences) emerge through STDP for synaptic connections between pyramidal cells in a model for generic
networks of pyramidal cells and PV+ inhibitory neurons that is based on data from the Petersen Lab (see
(Avermann et al, 2012). Shown are simulation results from (Pokorny et al., 2016) that corraborate earlier results
from (Klampfl and Maass, 2013) for a simplified model. b) Mean firing times of each neuron in one of the two
assemblies marked by a white dot, with a histogram of relative firing times in color coding on the same row. c) -e)
Emergent stereotypical network responses (d and e) to repeated input patterns (spoken digits, c) become
sufficiently different so that a WTA readout circuit of 4 neurons (bottom panel of d) learns without any supervision
to classify the spoken digits.
Constraint/Principle 3: Networks of neurons in the brain are spontaneously active and
have high trial-to-trial variability
Virtually all neural recordings show that network responses vary substantially from trial to
trial. This is no surprise, since not only channel kinetics in dendrites but also synaptic
transmission (Branco and Staras, 2005), (Kavalali, 2015) appear to be highly stochastic
processes. Furthermore networks of neurons in the brain represent ambiguous sensory
stimuli not through superimposed interpretations, but through spontaneous flips between
corresponding network states that represent alternative interpretations (Leopold and
Logothetis, 1999), (Jezek et al., 2011). Also spontaneous activity of networks of neurons
exhibits seemingly stochastic changes between highly structured network states, similary as
in response to stimuli (Luczak et al., 2009), (Berkes et al., 2011), (Miller et al., 2014).
These data force us to add stochasticity to the set of constraints for brain computation. Liquid
computing models, as discussed in the context of constraint 1, are compatible with noise. But
a key question is, whether models for computation with network states could benefit from
noise. In computational sciences there are in fact well known paradigms of computational
models that not only tolerate noise, but require noise for their computational strategy (Maass,
2014). For example, Markov chains are stochastic dynamical systems whose computational
function or program consists of their stationary distribution of network states, which they
approximate from any initial state after some finite mixing time. Boltzmann machines are
Markov chains that can be „embodied“ by networks of perceptrons with stochastic rules for
their switching between states 0 and 1. This class of computational models is good at solving
constraint satisfaction problems (Aarts and Korst, 1988), probabilistic inference through
Markov-chain Monte Carlo (MCMC) sampling (Andrieu et al., 2003), (Bishop, 2006) and for
learning generative models of complex distributions from examples..
One can also consider related stochastic models for brain computation, where recurrent
neural networks are viewed as Markov chains that sample from their stationary distribution of
network states, conditioned on the current input e to the network (termed evidence in the
language of probabilistic inference), see Fig. 3. It has been rigorously shown (Habenschuss
et al., 2013), that even quite realistic models for such networks with noise have a unique
stationary distribution p, where the firing or non-firing of each neuron is modelled by a
separate random variable. The inherent stochastic dynamics enables such model to estimate
∑ ,…,
, ,…, | ,
through MCMC sampling posterior marginals such as
where e could represent for example sensory evidence and internal goals, runs over all
possible values of random variables
that are irrelevant for the current computational taks,
and a binary variable
could represent for example the choice between two sources of
food. This posterior marginal can be estimated according to a sampling model by simply
observing the firing rate of a neuron whose firing or non-firing represents the values of the
. A network of neurons in the brain that exhibits for a choice task
binary random variable
fluctuating network states as proposed by MCMC-sampling has recently been discoved in
area OFC (Rich and Wallis, 2015).
Figure 3: Model for probabilistic inference through sampling in generic cortical microcircuits, see (Habenschuss
et al., 2013) for details. a) Synaptic weights and other parameters of a simple data-based generic cortical
microcircuit model encode a unque stationary distribution of network states. The network state at time t can be
defined for example as a binary vector that records which neuron fires in a small time window around t. Generic
circuit computation can be viewed as estimate of a posterior --conditioned on external input e-- through sampling
from the posterior distribution of network states. b) Instead of traditional measures for computation time, the time
needed to converge from an initial state to the stationary distribution of the network becomes relevant. Shown is a
common heuristic estimate (Gelman-Rubin ratio) for proximity of current sampling from the stationary distribution
(when the ratio enters the gray zone below 1.1 one usually says that convergence has taken place). Computer
simulations suggest quite fast convergence of estimates of a posterior marginal (solid lines: mean; dashed lines:
worst case) -independently of network size.
One interesting implication of sampling-based models for probabilistic inference is that one
has to re-analyze the concept of the required computing time (response time). A samplingbased model can provide a rough estimate very fast, but needs to look at more samples for a
more reliable answer. In addition, the samples that the stochastic system produces initially
are still influenced by its initial state. Hence the mixing time of a Markov chain is a key
component of the computing time for a stochastic computational model. According to first
simulation results this mixing time could lie for data-based models of cortical circuitry within
the range of behavioural response times (Fig. 3 c).
Theoretical results show that in principle every Boltzmann machine can be approximatly
represented through MCMC sampling of network states in a network of spiking neurons. The
term „neural sampling“ had been coined for the resulting model of brain computation
(Buesing et al, 2011). A suitable structure of cortical microcircuits enables networks of
spiking neurons in principle to even go beyond spike-based emulations of Boltzmann
machines with symmetric weights (which limits their capability to sample efficiently from
distributions with higher than 2nd order dependencies), and to represent (Pecevski et al.
2011) and learn (Pecevski and Maass, 2016) more general types of distributions over many
discrete variables in their stationary distribution of network states. Such distributions with
higher order dependencies are needed for example to include the „explaining away“ effect in
probabilistic inference that has been shown to be essential for visual perception (Knill and
Kersten, 1991). The hypothesis that the human brain encodes substantial amounts of
knowledge in the form of probability distributions has been subsequentially explored in
cognitive science (Tenenbaum et al., 2011).
Constraint/Principle 4: Networks configurations remain transient even in the adult
Experimental data show that local network connectivity (Holtmaat and Svoboda, 2009),
(Stettler et al, 2006), (Löwenstein at al., 2015) and neural codes (Ziv et al., 2013) are subject
to permanently ongoing changes, even in the adult brain. This constraint forces us to
consider the hypothesis that the brain samples not only network states (neural sampling) on
the fast time scale of producing a behavioural response, but also network configurations on
the slower time scale of learning (synaptic sampling). This hypothesis was recently explored
in (Kappel et al., 2015). The resulting model suggests that brain networks sample
continuously --through rewiring and synaptic plasticity-- from a stationary distribution of
network configurations (Fig. 4). This distribution has most of its probability mass on a lowdimensional submanifold of a very high-dimensional space that is distinguished through
structural constraints or priors (e.g., sparse connectivity), and good network performance.
The Fokker-Planck equation provides clear links between local stochastic rules for rewiring
and synaptic plasticity on one hand, and the resulting stationary distribution of network
configurations and parameter settings on the other hand. An immediate benefit of such
alternative organisation of network plasticity is that a network can immediately compensate
for internal or external changes, since no clever supervisor is needed to unlock a
permanently wandering network configuration from a previously found solution. Instead, such
compensation can be viewed as probabilistic inference on the basis of new evidence e, only
on the slower time scale of MCMC sampling from network configurations.
This model suggests that an approach to understand the structure and organization of brain
computations on the basis of compilation of connectivity data and neuron parameters from
many individuals faces a systematic problem: These data just represent snapshots from
somewhat unrelated dynamic processes in different individuals (Marder et al., 2006), and
one would need to collect and compare their statistics from large numbers of specimen.
Salient invariant properties and rules of neural circuits that are in permanent transition are
likely to be encoded more directly in the underlying genetic machinery.
Figure 4: Replacing the traditional maximum likelihood learning (ML) paradgm by sampling from a posterior of
network configurations. Upper 3 plots illustrate common ML learning paradigm for a neural network modelled as
generative model. The goal of learning is to reach a local optimum, which generally depends on the initial state of
the learning system. Lower 4 plots show integration of a prior on generally desirable network configurations and
suggest continuous sampling of network configurations from the resulting posterior through synaptic sampling
(see (Kappel et al., 2015) for details).
The four constraints/principles for models of brain computation that I have discussed are
compatible with each other. And they have to be compatible, since experimental tell us that
they are all --more or less-- present in networks of neurons in the brain. But obviously there
are tradeoffs between them from the functional perspective (see e.g. Fig. 12 in (Klampfl and
Maass, 2013) for a tradeoff between the first two principles). Hence one may conjecture that
the relative weight of each principle is regulated by the brain for each area and
developmental stage in a task dependent manner.
Altogether I have argued that the currently avaible experimental data provide useful guidance
for understanding how cognition and behaviour is implemented and regulated by networks of
neurons in the brain. A scientific approach that integrates experimental and theoretical
neuroscience looks like a straightforward strategy. But it may be seen as contradicting the
strategy proposed by (Marr and Poggio, 1976), to distinguish three levels of understanding
brain computations:
-- the computational (behavioural) level
-- the algorithmic level
-- the biological implementation level.
This partition, and an emergent focus on the upper two levels, was certainly fruitful a few
decades ago, when a good understanding on the computational level and functionally
powerful models on the algorithmic level were still missing. However in view of the
tremendous progress since then in algorithm design, theoretical neuroscience, and machine
learning, we are now faced with numerous competing models on the algorithmic level that
are all attractive from the functional perspective. The currently available experimental data
can help us now to select and refine these models on the basis of constraints from the
biological implementation level.
Acknowledgements: I would like to thank David Kappel, Robert Legenstein, and Zhaofei Yu
for helpful comments. This review was written under partial support the by the European
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