Section: New Results

Neural Networks as dynamical systems

Latching dynamics in neural networks with synaptic depression

Participants : Elif Köksal Ersöz, Carlos Aguilar [Université de Nice - BCL] , Pascal Chossat [Université de Nice - LJAD, Inria MathNeuro] , Martin Krupa [UCA, Inria MathNeuro] , Frédéric Lavigne [Université de Nice - BCL] .

Prediction is the ability of the brain to quickly activate a target concept in response to a related stimulus (prime). Experiments point to the existence of an overlap between the populations of the neurons coding for different stimuli, and other experiments show that prime-target relations arise in the process of long term memory formation. The classical modelling paradigm is that long term memories correspond to stable steady states of a Hopfield network with Hebbian connectivity. Experiments show that short term synaptic depression plays an important role in the processing of memories. This leads naturally to a computational model of priming, called latching dynamics; a stable state (prime) can become unstable and the system may converge to another transiently stable steady state (target). Hopfield network models of latching dynamics have been studied by means of numerical simulation, however the conditions for the existence of this dynamics have not been elucidated. In this work we use a combination of analytic and numerical approaches to confirm that latching dynamics can exist in the context of a symmetric Hebbian learning rule, however lacks robustness and imposes a number of biologically unrealistic restrictions on the model. In particular our work shows that the symmetry of the Hebbian rule is not an obstruction to the existence of latching dynamics, however fine tuning of the parameters of the model is needed.

This work has been published in PLoS one and is available as [13].

A natural follow-up of the work which has lead to the article [13] has been initiated through the postdoc project of Elif Köksal Ersöz. The objective is to extend the previous results in several ways. First, to gain more robustness in the heteroclinic chains sustained by the network model. Second, to be able to simulate much larger networks and exhibit heteroclinic dynamics in them. Third, to link with experimental data. The postdoc of Elif Köksal Ersöz is funded by the “tail” of the ERC Advanced Grant NerVi held by Olivier Faugeras.

Special issue for Martin Golubitsky

Participants : Pietro-Luciano Buono University Of Ontario Institute Of Technology, Canada ,, Martin Krupa [UCA, Inria MathNeuro] , Ian Stewart [University of Warwick, UK] .

The work is the introducion of this special issue, co-edited by Martin Krupa. It has been published in Dynamical Systems: An International Journal and is available as [17].

Consecutive and non-consecutive heteroclinic cycles in Hopfield networks

Participants : Pascal Chossat [Université de Nice - LJAD, Inria MathNeuro] , Martin Krupa [UCA, Inria MathNeuro] .

We review and extend the previous work [38] where a model was introduced for Hopfield-type neural networks, which allows for the existence of heteroclinic dynamics between steady patterns. This dynamics is a mathematical model of periodic or aperiodic switching between stored information items in the brain, in particular, in the context of sequential memory or cognitive tasks as observed in experiments. The basic question addressed in this work is whether, given a sequence of steady patterns, it is possible by applying classical learning rules to build a matrix of connections between neurons in the network, such that a heteroclinic dynamics links these patterns. It has been shown previously that the answer is positive in the case where the sequence is a so-called simple consecutive cycle. We show that on the contrary the answer is negative for a non-simple cycle: heteroclinic dynamics does still exist; however, it cannot follow the sequence of patterns from which the connectivity matrix was derived.

This work has been published in Dynamical Systems: An International Journal and is available as [21].

Asymptotic stability of pseudo-simple heteroclinic cycles in ${ℝ}^4$

Participants : Pascal Chossat [Université de Nice - LJAD, Inria MathNeuro] , Olga Podvigina [Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russia] .

Robust heteroclinic cycles in equivariant dynamical systems in ${ℝ}^4$ have been a subject of intense scientific investigation because, unlike heteroclinic cycles in ${ℝ}^3$, they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work [51], we have compiled an exhaustive list of finite subgroups of $O\left(4\right)$ admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyze the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in ${ℝ}^4$.

This work has been published in Journal of Nonlinear Science and is available as [26].

The period adding and incrementing bifurcations: from rotation theory to applications

Participants : Albert Granados [Polytechnic University of Catalonia, Barcelona, Spain] , Lluís Alsedà [Autonomous University of Barcelona, Spain] , Martin Krupa [UCA, Inria MathNeuro] .

This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and “rotation” numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the period incrementing bifurcation, in its proof relies on results for maps on the interval. We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasi-contractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.

This work has been published in SIAM Review and is available as [24].

Inverse correlation processing by neurons with active dendrites

Participants : Tomasz Górski [UNIC, CNRS, France] , Romain Veltz, Mathieu Galtier [UNIC, CNRS, France] , Helissande Fragnaud [UNIC, CNRS, France] , Bartosz Teleńczuk [UNIC, CNRS, France] , Alain Destexhe [UNIC, CNRS, France] .

In many neuron types, the dendrites contain a significant density of sodium channels and are capable of generating action potentials, but the significance and role of dendritic sodium spikes are unclear. Here, we use simplified computational models to investigate the functional effect of dendritic spikes. We found that one of the main features of neurons equipped with excitable dendrites is that the firing rate of the neuron measured at soma decreases with increasing input correlations, which is an inverse relation compared to passive dendrite and single-compartment models. We first show that in biophysical models the collision and annihilation of dendritic spikes causes an inverse dependence of firing rate on correlations. We then explore this in more detail using excitable dendrites modeled with integrate-and-fire type mechanisms. Finally, we show that the inverse correlation dependence can also be found in very simple models, where the dendrite is modeled as a discrete-state cellular automaton. We conclude that the cancellation of dendritic spikes is a generic mechanism that allows neurons to process correlations inversely compared to single-compartment models. This qualitative effect due to the presence of dendrites should have strong consequences at the network level, where networks of neurons with excitable dendrites may have fundamentally different properties than networks of point neuron models.

This work has been submitted for publication and is available as [33].