A Markovian event-based framework for stochastic spiking neural networks.

Abstract

In spiking neural networks, the information is conveyed by the spike times, that depend on the intrinsic dynamics of each neuron, the input they receive and on the connections between neurons. In this article we study the Markovian nature of the sequence of spike times in stochastic neural networks, and in particular the ability to deduce from a spike train the next spike time, and therefore produce a description of the network activity only based on the spike times regardless of the membrane potential process. To study this question in a rigorous manner, we introduce and study an event-based description of networks of noisy integrate-and-fire neurons, i.e. that is based on the computation of the spike times. We show that the firing times of the neurons in the networks constitute a Markov chain, whose transition probability is related to the probability distribution of the interspike interval of the neurons in the network. In the cases where the Markovian model can be developed, the transition probability is explicitly derived in such classical cases of neural networks as the linear integrate-and-fire neuron models with excitatory and inhibitory interactions, for different types of synapses, possibly featuring noisy synaptic integration, transmission delays and absolute and relative refractory period. This covers most of the cases that have been investigated in the event-based description of spiking deterministic neural networks.

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arXiv:0911.3462v2 [stat.AP] 18 Feb 2011
A Markovian event-based framework for stochastic spiking
neural networks
Jonathan Touboul1and Olivier Faugeras1
1NeuroMathComp Laboratory, INRIA, Sophia Antipolis, France
February 21, 2011
Abstract
In spiking neural networks, the information is conveyed by the spike times, that depend on the intrinsic
dynamics of each neuron, the input they receive and on the connections between neurons. In this article
we study the Markovian nature of the sequence of spike times in stochastic neural networks, and in
particular the ability to deduce from a spike train the next spike time, and therefore produce a description
of the network activity only based on the spike times regardless of the membrane potential process.
To study this question in a rigorous manner, we introduce and study an event-based description of
networks of noisy integrate-and-fire neurons, i.e. that is based on the computation of the spike times.
We show that the firing times of the neurons in the networks constitute a Markov chain, whose transition
probabilityis relatedtotheprobabilitydistributionoftheinterspikeintervaloftheneuronsinthenetwork.
In the cases where the Markovianmodel can be developed,the transition probability is explicitly derived
in such classical cases of neural networks as the linear integrate-and-fireneuron models with excitatory
and inhibitory interactions, for different types of synapses, possibly featuring noisy synaptic integration,
transmission delays and absolute and relative refractory period. This covers most of the cases that have
been investigated in the event-based description of spiking deterministic neural networks.
Introduction
Growing experimental evidence tends to establish that spike timings are essential to account for neu-
ral computations (see e.g. (Delorme and Thorpe, 2001; Fabre-Thorpe et al., 1998; Thorpe et al., 2001)).
This fact has motivated the use of spiking neuron models, rather than the traditional rate-based models,
and a very general interest for the characterization of the structure of spike times as well as for their
precise and efficient computation.
One of the great challenges in computational neuroscience consists in inferring from a sequence of
spike times the next spike that will be fired in the network. In other words, knowing the input a neuron
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receives and the network’s connectivitymap, can one computethe dynamics of the spike times regardless
of the actual value of the underlying membrane potential value? Answering this question amounts defin-
ing what is called an event-based description of the network (see e.g. the review of (Brette et al., 2007)).
The purpose of these models is to provide an exact simulation of the network, and this is precisely based
on the ability to infer the next spike time based on the knowledge of the previous spike times and on the
parameters of the models, without referring to the underlying membrane potential.
As discussed in (Brette et al., 2007), in the deterministic case, this property of event-based models is
true only for very few cases of simple types of neuron models, where explicit solutions to the membrane
potential voltage equation are available. Deterministic event-based algorithms were only developed for
simple pulse-coupledintegrate-and-firemodels (Watts, 1994; Claverol et al., 2002; Delorme and Thorpe,
2003), and more complex ones in the case of the Spike Response Model (Makino, 2003; Marian et al.,
2002; Gerstner and Kistler, 2002b). Rudolph and Destexhe developeda class of models that allow event-
based simulation (Rudolph and Destexhe, 2006) including a model of synaptic conductances. Romain
Brette described in (Brette, 2006, 2007) algorithms that apply to some classes of input current (exponen-
tial currents), and Tonnelierand collaboratorsextendedthese works to the case of the quadraticintegrate-
and-fire models in a case where the equation is integrable in closed-form (Tonnelier et al., 2007). Even
in this case the authors must make quite restrictive assumptions with regards to the connectivity and the
input current while being more general in terms of the intrinsic dynamics. Their approach relies on the
fact that the solutionof the particularequationtheychoose has a closed-formsolution. The priceto payis
that no generalization is possible. All these models share the property that the solution to the membrane-
potential equation can be found in closed-form, and therefore the spike times can be expressed as the
times of the first intersection of these solutions to the threshold, i.e. by inverting explicitly the solution.
Moreover, these models only apply to deterministic inputs of a very specific form (either constant, or
some types of exponential currents for the linear IF neuron).
Theyalsohavethecommonpropertythattheydonottakeintoaccountthenaturalrandomnesspresent
in the input to the neurons. In the cortex, noise constitutes a prominent aspect of the neural activity, and
is even sometimes assumed to have a functional role (see e.g. (Rolls and Deco, 2010)). A common
model for this random arrival of spikes consists in considering that the neuron receives a Brownian noisy
current (diffusion approximation), as we further explain in section 1. Taking into account the effect
of noise in the spike timings is a very difficult task, even for single neurons, and has been extensively
studied in the past decades and still constitutes an active area of research (see e.g. the books (Holden,
1976), (Ricciardi and Smith, 1977) and (Gerstner and Kistler, 2002b) or (Touboul and Faugeras, 2007)
and the references therein for a review).
In the stochastic case, event-based descriptions of spike timings become much more complicated.
Indeed, even when we have a closed-form expression of the membrane potential (which only occurs in
very few specific cases) there are at least two reasons why the problem is extremely difficult to solve.
First, it is not always possible to invert the probability density function of the membrane potential to find
the first crossing time to the threshold. Second is the fact that the spike times of all neurons can have a
very complexcorrelationstructure with the underlyingmembranepotential voltage: evenif we were able
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to compute the probability distribution of the next spike numerically, the dependence on the membrane
potential variable makes the next spike time extremely difficult to tame.
Extending these results to the statistics of spike times in a network constitutes therefore the next step
in both the theory of event-based descriptions of networks and of spike statistics. Indeed, the approach
proposedin the deterministic case of solving in closed-formthe membrane-potentialequationand invert-
ingit will fail. This problemwas alreadyidentifiedinthe reviewbyBrette andcollaborators(Brette et al.,
2007), where the authors mention that “the problem of simulating directly a stochastic process in asyn-
chronous algorithms is much harder because even for the simplest stochastic neuron models, there is no
closed analytical formula for the distribution of the time to the next spike (see e.g. (Tuckwell, 1988))”.
Anotherextremelycomplexproblemlies inthe dependencestructureofthe spiketimes amongthediffer-
ent neurons of the network, and the statistical correlation of the next spike time to the trajectory (sample
path) of the membrane potential. These are precisely the problems we address in the present manuscript.
Existing work on event-based descriptions of neuronal networks is based on a perfect match between
spike times of the voltage process and the event-based model. If this aim is highly relevant in the deter-
ministic case, in the stochastic case the relevant information is the probability distribution of the spike
times (i.e. the probability of a given neuron to fire at a given time) rather than some specific realizations
of spike trains. Our approach therefore aims at perfectly reproducing the probability distribution of the
spike times of the network.
In all the paper, we consider networks composed of N interacting integrate-and-fire spiking neurons
subjected to noisy inputs from the external environment. Mathematically, the membrane potential of
these neurons is defined by a stochastic process: each neuron receives at its synapses noisy inputs corre-
sponding to the random activity of ion channels and to the external activity of the network. During the
time intervals where no spike is emitted in the network, the membrane potential of each neuron evolves
independently according to its intrinsic dynamics. When the membrane potential V(i)(t) of the neuron
indexed by i reaches its deterministic threshold function θ(t) at time t0, the neuron elicits an action
potential. Subsequently, its membrane potential is reset to a given value V(i)
r , and the states of all its
postsynaptic neurons j are modified. We denote by V(i) the postsynaptic neighborhood of the neuron i,
i.e. the set of neurons that receive the spikes fired by neuron i. When a spike is transmitted to a neigh-
boring cell, it results in the emergence of a positive or negative postsynaptic pulse whose sign depends
on the connection type and whose shape depends on the neurotransmitters and the receptors.
In this article, we address the problem of event-based models for stochastic networks, and more
fundamentallyof the Markoviannature of the spike times, in some simple types of networks of integrate-
and-fire neurons with different intrinsic dynamics and different kinds of synaptic integration.To this
purpose, we introduce a special mathematical framework to describe the dynamics of the spikes in sec-
tion 1. We then start by addressing the case of purely inhibitory networks, and we show in section 2 that
the spike times can be described throughthe introduction of a Markov chain whose transition probability
is related to a first passage-time problem, for some simple types of neuron models. We further extend
this approach in appendix A to more complex bio-inspired neuronal descriptions, and show the Markov
property does not necessarily hold true in all cases. The more slightly complex case of networks includ-
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ing excitatory interactions is then treated in section 3, and the same conclusion holds true: a Markov
chain can be defined in relation with the spike times in the network, whose transitions are related to first
passage time problems. The present framework and derivations open the way to different theoretical,
numerical and computational developments, some of which are discussed in the conclusion.
1Theoretical framework
We start byintroducingthegeneralmathematicalconceptsandtheframeworkthatwe will beusingin this
manuscript in order to study spike times in stochastic integrate-and-fire networks. The first and central
concept we want to recall is the Markov property for a random sequence. Basically, assume that the
random sequence (Xk)k∈Nis defined by a certain recurrence property given by the transition probability
density p(Xn+1|Xn, ..., X0). This expresses probability density of the process Xn+1conditionally to
the values of X0,...,Xn. The random sequence is called a Markov chain if this conditional probability
depends only on Xn, being independent of (Xk)k=0,...n−1, that is:
p(Xn+1= xn+1|Xn= xn, ..., X0= x0) = p(Xn+1= xn+1|Xn= xn).
Loosely speaking, the knowledge of the present state makes the future of the process independent of its
past states.
The building block of our framework consists in considering the dynamics of the spike times instead
of the dynamics of all variables involved in the description of the neuron’s state. This purpose motivates
the introduction of what we call the countdown process, a stochastic process governing the dynamics of
the spikes in the network and defined as follows:
Definition 1.1. [Countdown process] For each neuron i, we define X(i)(t) ≥ 0 to be the remaining time
until the next emission of a spike by neuron i if it does not receive any spike meanwhile. We call this
stochastic process the countdown process of the neuron i.
stochastic calculus literature, we also sometimes note X(i)
Xt= (X(1)
t
) is called the countdown process of the network, or simply countdown process.
Following the tradition established in the
t
for X(i)(t).The N dimensional process
t ,...,X(N)
For each neuron i ∈ {1,...,N}, the dynamics of (X(i)(t)) is very simple, linearly decreasing with
slope −1 during the intervals of time where no spike is received or produced:
dX(i)(t)
dt
= −1
(1)
At time t, the next spike will be fired by neuron i = ArgMinj∈1...NX(j)(t) at time t + X(i)(t).
At spike time, the membrane potential of the neuron that just fired an action potential is reset, and
therefore the new countdown value after the spike emission is updated to a new value corresponding to
the next spike time of this neuron if nothing occurs meanwhile. This random variable, noted Y(i), is
referred to as the reset random variable.
Each neuron connected to the one that fired the action potential has his state modified, and its next
firing time is reset accordingly, conditionally to the value of the next firing time predicted before the
reception of the spike.
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Figure1: Arepresentationofasamplepathforthecountdownprocessandtherelatedmembranepotential
(case of the perfect integrate-and-fireneurons network).
The next spike time of neuron j is modified consequently to the reception of the spike from neuron
i and the new countdown value of neuron j after spike emission is reset according to a random variable
denoted ηij, corresponds to the next firing time of neuron j after taking into account the reception of
the incoming spike from i, and conditionally on the countdown value at the spike reception (see Fig
1)1. This random variable is called the interaction random variable.We observe that the countdown
process is a piecewise deterministic process such as the ones described in (Davis, 1984). The spike
times (t1,...,tn,...) are the only possible discontinuities of the countdown process. The countdown
sequence is defined as the value of the N-dimensional countdown process of the network sampled just
after the spike emission: Xn:= X(t+
is equivalent to the knowledge of the countdown sequence, as we now prove.
n). It is easy to show that the knowledge of the countdown process
Proposition 1.1. There is a path-wise bi-univocal correspondence between the continuous-time count-
down process and the discrete-time countdown sequence.
Proof. Assumethatwehavearealizationofthecountdownprocess(Xt)t∈[0,T]. Thespiketimes(t1,...,tn,...)
are given by the zeros of the process, and the value of the process just after these spike times Xn :=
1and possibly on different other variables depending on the model considered.
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