Mathematical analysis of a neural network with inhibitory coupling

Use of an hourglass model in neuronal coding

We study a system of interacting renewal processes which is a model for neuronal activity. We show that the system possesses an exponentially large number (with respect to the number of neurons in the network) of limiting configurations of the ‘firing neurons’. These we call patterns. Furthermore, under certain conditions of symmetry we find an algorithm to control limiting patterns by means of the connection parameters.

Stationarity properties of neural networks

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y (i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y (i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Stationarity properties of neural networks

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Results on a binding neuron model and their implications for modified hourglass model for neuronal network

The classical models of single neuron like Hodgkin-Huxley point neuron or leaky integrate and fire neuron assume the influence of postsynaptic potentials to last till the neuron fires. Vidybida (2008) in a refreshing departure has proposed models for binding neurons in which the trace of an input is remembered only for a finite fixed period of time after which it is forgotten. The binding neurons conform to the behaviour of real neurons and are applicable in constructing fast recurrent networks for computer modeling. This paper develops explicitly several useful results for a binding neuron like the firing time distribution and other statistical characteristics. We also discuss the applicability of the developed results in constructing a modified hourglass network model in which there are interconnected neurons with excitatory as well as inhibitory inputs. Limited simulation results of the hourglass network are presented.

A Markovian event-based framework for stochastic spiking neural networks

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.

Study of synaptic plasticity via random graphs

The dynamical random graphs associated with a certain class of biological neural networks are introduced and studied. We describe the phase diagram revealing the parameters of a single neuron and of the synaptic strengths which allow formation of the stable strongly connected large groups of neurons. It is shown that the cycles are the most stable structures when the Hebb rule is implemented into the dynamics of the network of excitatory neurons. We discuss the role of cycles for the synchronization of the neuronal activity.

Neural networks through the hourglass

The effect of the synaptic plasticity on the dynamics of a large neural network is studied. Our approach is analytic but inspired by the data, both simulated and experimental. We explain formation of the small strongly connected assemblies within a dynamical network following Hebb's rule. Also, we find the conditions for the synchrony effect in the stochastic network in the absence of large synchronized input.

Use of an hourglass model in neuronal coding

We study a system of interacting renewal processes which is a model for neuronal activity. We show that the system possesses an exponentially large number (with respect to the number of neurons in the network) of limiting configurations of the ‘firing neurons’. These we call patterns. Furthermore, under certain conditions of symmetry we find an algorithm to control limiting patterns by means of the connection parameters.

Stochastic dynamics of a neural network with inhibitory and excitatory connections

We analyze the dynamics of a stochastic neural model. We show that an inhibitory neuron incorporated into a population of connected excitatory neurons, stabilizes the oscillations of the total activity.

Effects of collateral inhibition in a model of the immature rat cerebellar cortex: multineuron correlations

A model of the immature rat cerebellar cortex is used to simulate the effect of the inhibitory recurrent collateral axons of the Purkinje cells on the spike trains in the network. Inhibition induces an important overall change in the statistical characteristics of individual spike trains. It is also instrumental in producing a strong cooperativity between the different neurons. Moreover, a functional spatial anisotropy appears. A specific entropy index is used to analyze levels of information transfer between clustered and faraway neurons in the network. The formatting effect of recurrent collateral inhibition on spike trains and on network functional dynamics is studied by means of a model of the newborn rat cerebellar cortex. This immature structure has simpler morphological characteristics and fewer physiological parameters than the adult one. It is thus a good candidate for the comparison between experimental and theoretical data. The model network is made of 256 formal neurons (FN), arranged in a square lattice. Each neuron is coupled to its eight nearest neighbors by inhibitory links. All the parameters of the different elements of the model--in particular integration of inhibitory and excitatory inputs--are given anatomical and physiological values derived from biological data. Activities of single FNs and correlations between spatially distant ones are analyzed with classical statistical techniques as well as with a specific informational entropy method we introduce. Simulation results indicate that inhibition is instrumental in: (1) the transformation of the spike train characteristics. This includes a lengthening of the mean interspike interval as well as an overall change in the statistical distribution of intervals, with an emergence of long-lasting ones; (2) the functional structuration of the network. Inhibitory connections between nearest neighbors induce a strong cooperativity between FNs. Furthermore a clear spatial anisotropy occurs in the functioning of the network, with inhibitory effects extending beyond local connectivity in preferential directions. We propose an interpretation of this functional structuration in terms of the various routes followed by the inhibition, including relay effects. The parameters of the model (levels of activities, inhibition rules and connectivities) were varied in order to test the robustness of the above results. Finally, the results are compared with those obtained in an experimental situation.