Vladimir Klinshov

Dense Neuron Clustering Explains Connectivity Statistics in Cortical Microcircuits

Local cortical circuits appear highly non-random, but the underlying connectivity rule remains elusive. Here, we analyze experimental data observed in layer 5 of rat neocortex and suggest a model for connectivity from which emerge essential observed non-random features of both wiring and weighting. These features include lognormal distributions of synaptic connection strength, anatomical clustering, and strong correlations between clustering and connection strength. Our model predicts that cortical microcircuits contain large groups of densely connected neurons which we call clusters. We show that such a cluster contains about one fifth of all excitatory neurons of a circuit which are very densely connected with stronger than average synapses. We demonstrate that such clustering plays an important role in the network dynamics, namely, it creates bistable neural spiking in small cortical circuits. Furthermore, introducing local clustering in large-scale networks leads to the emergence of various patterns of persistent local activity in an ongoing network activity. Thus, our results may bridge a gap between anatomical structure and persistent activity observed during working memory and other cognitive processes.

Activity clusters in dynamical model of the working memory system.

A mathematical model of working memory is proposed in the form of a network of neuron-like units interacting via global inhibitory feedback. This network is capable of storing information items in the form of clusters of periodical spiking activity. Several sequentially excited clusters can coexist simultaneously, corresponding to several items stored in the memory. The capacity of the memory is studied as the function of the system parameters.

WORKING MEMORY IN THE NETWORK OF NEURON-LIKE UNITS WITH NOISE

A model of working memory system in the form of a network of neuron-like units is considered. The units are stimulated by a periodic subthreshold stimulus and interact via inhibitory feedback. The information storage in the system occurs in the form of periodic clusters of neural activity. Several sequentially firing clusters can coexist in the network simultaneously corresponding to several stored images. The influence of noise on cluster dynamics and noise resistance of the system are studied. Optimal parameters corresponding to maximal noise resistance of the system were found.

Phase response curves for models of earthquake fault dynamics.

We systematically study effects of external perturbations on models describing earthquake fault dynamics. The latter are based on the framework of the Burridge-Knopoff spring-block system, including the cases of a simple mono-block fault, as well as the paradigmatic complex faults made up of two identical or distinct blocks. The blocks exhibit relaxation oscillations, which are representative for the stick-slip behavior typical for earthquake dynamics. Our analysis is carried out by determining the phase response curves of first and second order. For a mono-block fault, we consider the impact of a single and two successive pulse perturbations, further demonstrating how the profile of phase response curves depends on the fault parameters. For a homogeneous two-block fault, our focus is on the scenario where each of the blocks is influenced by a single pulse, whereas for heterogeneous faults, we analyze how the response of the system depends on whether the stimulus is applied to the block having a shorter or a longer oscillation period.

Phase response function for oscillators with strong forcing or coupling

Phase response curve (PRC) is an extremely useful tool for studying the response of oscillatory systems, e.g. neurons, to sparse or weak stimulation. Here we develop a framework for studying the response to a series of pulses which are frequent or/and strong so that the standard PRC fails. We show that in this case, the phase shift caused by each pulse depends on the history of several previous pulses. We call the corresponding function which measures this shift the phase response function (PRF). As a result of the introduction of the PRF, a variety of oscillatory systems with pulse interaction, such as neural systems, can be reduced to phase systems. The main assumption of the classical PRC model, i.e. that the effect of the stimulus vanishes before the next one arrives, is no longer a restriction in our approach. However, as a result of the phase reduction, the system acquires memory, which is not just a technical nuisance but an intrinsic property relevant to strong stimulation. We illustrate the PRF approach by its application to various systems, such as Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF allows predicting the dynamics of forced and coupled oscillators even when the PRC fails.

Slow rate fluctuations in a network of noisy neurons with coupling delay

We analyze the emergence of slow rate fluctuations and rate oscillations in a model of a random neuronal network, underpinning the individual roles and interplay of external and internal noise, as well as the coupling delay. We use the second-order finite-size mean-field model to gain insight into the relevant parameter domains and the mechanisms behind the phenomena. In the delay-free case, we find an intriguing paradigm for slow stochastic fluctuations between the two stationary states, which is shown to be associated to noise-induced transitions in a double-well potential. While the basic effect of coupling delay consists in inducing oscillations of mean rate, the coaction with external noise is demonstrated to lead to stochastic fluctuations between the different oscillatory regimes.

Multi-jittering Instability in Oscillatory Systems with Pulse Coupling

In oscillatory systems with pulse coupling regular spiking regimes may destabilize via a peculiar scenario called “multi-jitter instability”. At the bifurcation point numerous so-called “jittering” regimes with distinct inter-spike intervals emerge simultaneously. Such regimes were first discovered in a single oscillator with delayed pulse feedback and later were found in networks of coupled oscillators. The present chapter reviews recent results on multi-jitter instability and discussed its features.

Dynamics of Oscillatory Networks with Pulse Delayed Coupling

The chapter is devoted to the dynamics of networks of oscillators with pulse time-delayed coupling. We develop a mathematical technique that allows to reduce the dynamics of such networks to multi-dimensional maps. With the help of these maps we consider networks of various configurations: a single oscillator with feedback, a feed-forward ring, a pair of oscillators with mutual coupling, a small network with heterogeneous delays, and a large network with all-to-all coupling. In all these examples we show that the role of the delay is significant and leads to the modification of the existing dynamical regimes and the emergence of new ones.

Clustering promotes switching dynamics in networks of noisy neurons

Macroscopic variability is an emergent property of neural networks, typically manifested in spontaneous switching between the episodes of elevated neuronal activity and the quiescent episodes. We investigate the conditions that facilitate switching dynamics, focusing on the interplay between the different sources of noise and heterogeneity of the network topology. We consider clustered networks of rate-based neurons subjected to external and intrinsic noise and derive an effective model where the network dynamics is described by a set of coupled second-order stochastic mean-field systems representing each of the clusters. The model provides an insight into the different contributions to effective macroscopic noise and qualitatively indicates the parameter domains where switching dynamics may occur. By analyzing the mean-field model in the thermodynamic limit, we demonstrate that clustering promotes multistability, which gives rise to switching dynamics in a considerably wider parameter region compared to the case of a non-clustered network with sparse random connection topology.

Embedding the dynamics of a single delay system into a feed-forward ring

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example, we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.