Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Abstract

PACS 05.45.Ac -Low-dimensional chaos PACS 05.45.Pq -Numerical simulations of chaotic systems PACS 87.19.ll -Models of single neurons and networks Abstract -We characterize the systematic changes in the topological structure of chaotic attractors that occur as spike-adding and homoclinic bifurcations are encountered in the slow-fast dynamics of neuron models. This phenomenon is detailed in the simple Hindmarsh-Rose neuron model, where we show that the unstable periodic orbits appearing after each spike-adding bifurcation are associated with specific symbolic sequences in the canonical symbolic encoding of the dynamics of the system. This allows us to understand how these bifurcations modify the internal structure of the chaotic attractors.The wide-range assessment of brain and behaviors is one of the pivotal challenges of this century. To understand how an incredibly sophisticated system such as the brain per se functions dynamically, it is imperative to study the dynamics of its constitutive elements -neurons. Therefore, the design of mathematical models for neurons has arisen as a trending topic in science for a few decades, since Hodgkin and Huxley developed the first model of action potentials in the neuron membrane [1]. Starting from that seminal mathematical model, a lot of variant models describing different kinds of neuron cells in numerous animals have been proposed in the literature. For instance, a reduced model [2-4] of the bursting of leech heart neuron is specified by the following three equations derived through the Hodgkin-Huxley gated variable formalism: As control parameters of the system are varied, its dynamics undergoes bifurcations such as classified by dynamical systems theory that match qualitative metamorphoses in terms specific to neuroscience. A broad range of non-stationary activity types can be observed, which includes regular and irregular tonic spiking, bursting and mixed-mode oscillations and combinations of them, as well as oscillatory transients toward quiescent states. In terms of dynamical system theory, these behaviors correspond to stable periodic and deterministically chaotic orbits in the phase space of the model. Figure 1 represents a twoparameter sweep of the leech model in the (τ K2 , I app )-plane, where the number of spikes per period, measured with the spike-counting method (SPC) [5], is color-coded. Banded structures correlated with zones of chaotic behavior appear clearly in this diagram. They are associated with spike-adding bifurcations, where the number of spikes is incremented by one, as indicated in the figure.Spike-adding bifurcations are special bifurcations that are common in fast-slow systems. They lead to the appearance of extra spikes (turns) in the fast manifold region and are quite important in that they progressively p-1

A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

“…The results provide strong evidence that an individual neuron is capable of generating different bifurcation scenarios, which forms a framework for identifying the relationships between different firing patterns and different bifurcation scenarios in an isolated CCI model. The results of the present paper and other investigations (González-Miranda 2012; Barrio and Shilnikov 2011;Barrio et al 2015Barrio et al , 2014 provide deep and comprehensive insight into the nonlinear dynamics or bifurcations of neural firing patterns in a two-dimensional parameter space. Earlier studies observed bifurcation and chaos in neural firing patterns on axons and neurons stimulated by external signals (Hayashi et al 1982;Aihara et al 1984).…”

“…Many complex or novel nonlinear behaviors related to chaos or bifurcations have been investigated in the theoretical models, which are helpful for the progresses of nonlinear dynamics in neuroscience (González-Miranda 2005, 2012; Barrio and Shilnikov 2011;Barrio et al 2015Barrio et al , 2014Zheng and Tonnelier 2009;Yamada and Kashimori 2013;Wei et al 2014;Ma and Tang 2015).…”

Control strategies of 3-cell Central Pattern Generator via global stimuli

“…This method allows us to identify stability windows with constant spike numbers, as well as to detect the borders where the spike number changes. We can see these structures separated by spike-adding bifurcations 30 31 in Fig. 2 with clearly demarcated regions corresponding to bursting, tonic spiking and quiescence states.…”

Cascades of Global Bifurcations and Chaos near a Homoclinic Flip Bifurcation: A Case Study

“…Parameter sweeping techniques have been used in part to understand the nature of cascades of homoclinic bifurcation close to Bykov T-points in the Lorenz system [5] and Shimizu-Morioka system [5,42]. It is also useful to illustrate spike adding in neuron models; for example, see [3,4]. By combining continuation and parameter sweeping, we are able to characterize different phenomena not only in the vicinity of the homoclinic flip bifurcation point of case C, but also far away from it.…”