On Stationary Distributions of Stochastic Neural Networks

Abstract

The paper deals with nonlinear Poisson neuron network models with bounded memory dynamics, which can include both Hebbian learning mechanisms and refractory periods. The state of the network is described by the times elapsed since its neurons fired within the post-synaptic transfer kernel memory span, and the current strengths of synaptic connections, the state spaces of our models being hierarchies of finitedimensional components. We prove the ergodicity of the stochastic processes describing the behaviour of the networks, establish the existence of continuously differentiable stationary distribution densities (with respect to the Lebesgue measures of corresponding dimensionality) on the components of the state space, and find upper bounds for them. For the density components, we derive a system of differential equations that can be solved in a few simplest cases only. Approaches to approximate computation of the stationary density are discussed. One approach is to reduce the dimensionality of the problem by modifying the network so that each neuron cannot fire if the number of spikes it emitted within the post-synaptic transfer kernel memory span reaches a given threshold. We show that the stationary distribution of this 'truncated' network converges to that of the unrestricted network as the threshold increases, and that the convergence is at a superexponential rate. A complementary approach uses discrete Markov chain approximations to the network process.The anatomy of a neuron involves three distinct parts with different electrical activity functions: dendrites that form a tree and contain post-synaptic receptors (inputs), the cell body (soma) that integrates the input currents coming from the dendrites, and a long-limbed axon that terminates with pre-synaptic buttons (outputs). A typical feature of the neuronal electrical activity is the propagation of membrane depolarisation. The membrane of a resting neuron is polarised. Brief high-amplitude depolarisations that propagate from the soma along the axon are called action potentials (or spikes) and have a characteristic shape. When a spike reaches an axonal termination that 'connects' to a post-synaptic neuron, neurotransmitters are released into the extra cellular space and excite receptors on the post-synaptic neuron (usually on dendrites). This generates a local variation of the membrane potential in that neuron, which propagates towards the soma. The soma can be seen as a spatiotemporal integrator of these post-synaptic potentials (PSPs) to generate an output spike. The soma potential often remains close to the resting value for a few milliseconds after firing an action potential, which is referred to as the refractory period (during which the neuron cannot fire again). These basic elements of the neuronal information processing actually depend upon many different mechanisms at the molecular level, such as ionic concentrations, density of ion channels, axonal myelination, and types of neurotransmitters (for a review, we refer the reader to [1]).There exis...

Stochastic Computations in Cortical Microcircuit Models

“…The same mathematical framework also guarantees exponentially fast convergence to a stationary distribution of trajectories of network states (of any fixed time length), thereby providing a theoretical foundation for understanding stochastic computations with experimentally observed stereotypical trajectories of network states. These results extend and generalize previous work in [17] and [18] in two ways. First, previous convergence proofs had been given only for networks of simplified neurons in which the (sub-threshold) neuronal integration of pre-synaptic spikes was assumed a linear process, thereby excluding the potential effects of dendritic non-linearities or synaptic short-term dynamics.…”

“…In network models and experimental setups where slower processes significantly influence (or interfere with) the dynamics on shorter time scales, it would make sense to extend the concept of a stationary distribution to include, for example, also the synaptic parameters as random variables. A first step in this direction has been made for neurons with linear sub-threshold dynamics and discretized synapses in [18] .…”

“…We denote the space of all possible network states of length by or, when unambiguous, simply by . Note that this definition is equivalent to the state definition in [18] , to which the interested reader is referred for further formal details (e.g. the associated -algebra of the state space ).…”

“…Hence with at least probability no neuron fires within time units (cf. [18] ). In technical terms, this implies that the stochastic kernel corresponding to a duration of length fulfills the Doeblin condition [104] – a property which is highly useful for proving convergence and ergodicity results.…”

“…To this end, we prove here that even biologically realistic models for networks of neurons in the brain have – for a suitable definition of network state – a unique stationary distribution of network states. Previous work had focused in this context on neuronal models with linear sub-threshold dynamics [17] , [18] and constant external input (e.g. constant input firing rates).…”