Germán Mato

On numerical simulations of integrate-and-fire neural networks

It is shown that very small time steps are required to reproduce correctly the synchronization properties of large networks of integrate-and-fire neurons when the differential system describing their dynamics is integrated with the standard Euler or second-order Runge-Kutta algorithms. The reason for that behavior is analyzed, and a simple improvement of these algorithms is proposed.

Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons

We investigate theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory. The neurons are modeled with quadratic integrate-and-fire dynamics, which provide a very good approximation for the subthreshold behavior of a large class of neurons. In addition to their synaptic recurrent inputs, the neurons receive a tonic external input that varies from neuron to neuron. Because of its relative simplicity, this model can be studied analytically. We investigate the stability of the asynchronous state (AS) of the network with given average firing rates of the two populations. First, we show that the AS can remain stable even if the synaptic couplings are strong. Then we investigate the conditions under which this state can be destabilized. We show that this can happen in four generic ways. The first is a saddle-node bifurcation, which leads to another state with different average firing rates. This bifurcation, which occurs for strong enough recurrent excitation, does not correspond to the emergence of synchrony. In contrast, in the three other instability mechanisms, Hopf bifurcations, which correspond to the emergence of oscillatory synchronous activity, occur. We show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations. We also show that besides these codimension 1 bifurcations, the system can display several codimension 2 bifurcations: Takens-Bogdanov, Gavrielov-Guckenheimer, and double Hopf bifurcations.

The Combined Effects of Inhibitory and Electrical Synapses in Synchrony

Recent experimental results have shown that GABAergic interneurons in the central nervous system are frequently connected via electrical synapses. Hence, depending on the area or the subpopulation, interneurons interact via inhibitory synapses or electrical synapses alone or via both types of interactions. The theoretical work presented here addresses the significance of these different modes of interactions for the interneuron networks dynamics. We consider the simplest system in which this issue can be investigated in models or in experiments: a pair of neurons, interacting via electrical synapses, inhibitory synapses, or both, and activated by the injection of a noisy external current. Assuming that the couplings and the noise are weak, we derive an analytical expression relating the cross-correlation (CC) of the activity of the two neurons to the phase response function of the neurons. When electrical and inhibitory interactions are not too strong, they combine their effect in a linear manner. In this regime, the effect of electrical and inhibitory interactions when combined can be deduced knowing the effects of each of the interactions separately. As a consequence, depending on intrinsic neuronal proper-ties, electrical and inhibitory synapses may cooperate, both promoting synchrony, or may compete, with one promoting synchrony while the other impedes it. In contrast, for sufficiently strong couplings, the two types of synapses combine in a nonlinear fashion. Remarkably, we find that in this regime, combining electrical synapses with inhibition ampli-fies synchrony, whereas electrical synapses alone would desynchronize the activity of the neurons. We apply our theory to predict how the shape of the CC of two neurons changes as a function of ionic channel conduc-tances, focusing on the effect of persistent sodium conductance, of the firing rate of the neurons and the nature and the strength of their interac-tions. These predictions may be tested using dynamic clamp techniques.

Self-Similarity Properties of Natural Images Resemble Those of Turbulent Flows

We show that the statistics of an edge type variable in natural images exhibits self-similarity properties which resemble those of local energy dissipation in turbulent flows. Our results show that selfsimilarity and extended self-similarity hold remarkably for the statistics of the local edge variance, and that the very same models can be used to predict all of the associated exponents. These results suggest using natural images as a laboratory for testing more elaborate scaling models of interest for the statistical description of turbulent flows. The properties we have exhibited are relevant for the modeling of the early visual system: They should be included in models designed for the prediction of receptive fields. [S0031-9007(97)05190-9]

Synchrony in Heterogeneous Networks of Spiking Neurons

The emergence of synchrony in the activity of large, heterogeneous networks of spiking neurons is investigated. We define the robustness of synchrony by the critical disorder at which the asynchronous state becomes linearly unstable. We show that at low firing rates, synchrony is more robust in excitatory networks than in inhibitory networks, but excitatory networks cannot display any synchrony when the average firing rate becomes too high. We introduce a new regime where all inputs, external and internal, are strong and have opposite effects that cancel each other when averaged. In this regime, the robustness of synchrony is strongly enhanced, and robust synchrony can be achieved at a high firing rate in inhibitory networks. On the other hand, in excitatory networks, synchrony remains limited in frequency due to the intrinsic instability of strong recurrent excitation.

Short-Term Plasticity Explains Irregular Persistent Activity in Working Memory Tasks

Persistent activity in cortex is the neural correlate of working memory (WM). In persistent activity, spike trains are highly irregular, even more than in baseline. This seemingly innocuous feature challenges our current understanding of the synaptic mechanisms underlying WM. Here we argue that in WM the prefrontal cortex (PFC) operates in a regime of balanced excitation and inhibition and that the observed temporal irregularity reflects this regime. We show that this requires that nonlinearities underlying the persistent activity are primarily in the neuronal interactions between PFC neurons. We also show that short-term synaptic facilitation can be the physiological substrate of these nonlinearities and that the resulting mechanism of balanced persistent activity is robust, in particular with respect to changes in the connectivity. As an example, we put forward a computational model of the PFC circuit involved in oculomotor delayed response task. The novelty of this model is that recurrent excitatory synapses are facilitating. We demonstrate that this model displays direction-selective persistent activity. We find that, even though the memory eventually degrades because of the heterogeneities, it can be stored for several seconds for plausible network size and connectivity. This model accounts for a large number of experimental findings, such as the findings that have shown that firing is more irregular during the persistent state than during baseline, that the neuronal responses are very diverse, and that the preferred directions during cue and delay periods are strongly correlated but tuning widths are not.

Type i and type ii neuron models are selectively driven by differential stimulus features

Neurons in the nervous system exhibit an outstanding variety of morphological and physiological properties. However, close to threshold, this remarkable richness may be grouped succinctly into two basic types of excitability, often referred to as type I and type II. The dynamical traits of these two neuron types have been extensively characterized. It would be interesting, however, to understand the information-processing consequences of their dynamical properties. To that end, here we determine the differences between the stimulus features inducing firing in type I and type II neurons. We work with both realistic conductance-based models and minimal normal forms. We conclude that type I neurons fire in response to scale-free depolarizing stimuli. Type II neurons, instead, are most efficiently driven by input stimuli containing both depolarizing and hyperpolarizing phases, with significant power in the frequency band corresponding to the intrinsic frequencies of the cell.

Inhibition Potentiates the Synchronizing Action of Electrical Synapses

In vivo and in vitro experimental studies have found that blocking electrical interactions connecting GABAergic interneurons reduces oscillatory activity in the γ range in cortex. However, recent theoretical works have shown that the ability of electrical synapses to promote or impede synchrony, when alone, depends on their location on the dendritic tree of the neurons, the intrinsic properties of the neurons and the connectivity of the network. The goal of the present paper is to show that this versatility in the synchronizing ability of electrical synapses is greatly reduced when the neurons also interact via inhibition. To this end, we study a model network comprising two-compartment conductance-based neurons interacting with both types of synapses. We investigate the effect of electrical synapses on the dynamical state of the network as a function of the strength of the inhibition. We find that for weak inhibition, electrical synapses reinforce inhibition-generated synchrony only if they promote synchrony when they are alone. In contrast, when inhibition is sufficiently strong, electrical synapses improve synchrony even if when acting alone they would stabilize asynchronous firing. We clarify the mechanism underlying this cooperative interplay between electrical and inhibitory synapses. We show that it is relevant in two physiologically observed regimes: spike-to-spike synchrony, where neurons fire at almost every cycle of the population oscillations, and stochastic synchrony, where neurons fire irregularly and at a rate which is substantially lower than the frequency of the global population rhythm.

The interplay of seven subthreshold conductances controls the resting membrane potential and the oscillatory behavior of thalamocortical neurons

The signaling properties of thalamocortical (TC) neurons depend on the diversity of ion conductance mechanisms that underlie their rich membrane behavior at subthreshold potentials. Using patch-clamp recordings of TC neurons in brain slices from mice and a realistic conductance-based computational model, we characterized seven subthreshold ion currents of TC neurons and quantified their individual contributions to the total steady-state conductance at levels below tonic firing threshold. We then used the TC neuron model to show that the resting membrane potential results from the interplay of several inward and outward currents over a background provided by the potassium and sodium leak currents. The steady-state conductances of depolarizing Ih (hyperpolarization-activated cationic current), IT (low-threshold calcium current), and INaP (persistent sodium current) move the membrane potential away from the reversal potential of the leak conductances. This depolarization is counteracted in turn by the hyperpolarizing steady-state current of IA (fast transient A-type potassium current) and IKir (inwardly rectifying potassium current). Using the computational model, we have shown that single parameter variations compatible with physiological or pathological modulation promote burst firing periodicity. The balance between three amplifying variables (activation of IT, activation of INaP, and activation of IKir) and three recovering variables (inactivation of IT, activation of IA, and activation of Ih) determines the propensity, or lack thereof, of repetitive burst firing of TC neurons. We also have determined the specific roles that each of these variables have during the intrinsic oscillation.

Generalization properties of multilayered neural networks

Generalization properties of multilayered neural networks with binary couplings are studied in the high-temperature limit. The transition to the perfect generalization phase is evaluated for systems with an arbitrary number of layers. It is found that the thermodynamic transition occurs for a number of examples lower than for the perceptron, but the opposite occurs for the transition in which the poor generalization solution disappears. The generalization error is also decomposed according to the contributions coming from different numbers of hidden neurons that have a wrong sign in the internal field. This allows the authors to describe the generalization behaviour of the hidden neurons.