Research paper
Conditional Lyapunov exponents and transfer entropy in coupled bursting neurons under excitation and coupling mismatch

https://doi.org/10.1016/j.cnsns.2017.08.022Get rights and content

Highlights

  • A new method for computing conditional Lyapunov exponent is proposed.

  • The synchronism of Hindmarsh-Rose (HR) neurons is analyzed from different perspectives.

  • The stability and information flux are evaluated for coupled HR neurons under different scenarios.

  • The results reveal a particular structure on the synchronization region, how it grows under bi-directional coupling and robustness under excitation shift.

Abstract

This work has a twofold aim: (a) to analyze an alternative approach for computing the conditional Lyapunov exponent (λcmax) aiming to evaluate the synchronization stability between nonlinear oscillators without solving the classical variational equations for the synchronization error dynamical system. In this first framework, an analytic reference value for λcmax is also provided in the context of Duffing master-slave scenario and precisely evaluated by the proposed numerical approach; (b) to apply this technique to the study of synchronization stability in chaotic Hindmarsh-Rose (HR) neuronal models under uni- and bi-directional resistive coupling and different excitation bias, which also considered the root mean square synchronization error, information theoretic measures and asymmetric transfer entropy in order to offer a better insight of the synchronization phenomenon. In particular, statistical and information theoretical measures were able to capture similarity increase between the neuronal oscillators just after a critical coupling value in accordance to the largest conditional Lyapunov exponent behavior. On the other hand, transfer entropy was able to detect neuronal emitter influence even in a weak coupling condition, i.e. under the increase of conditional Lyapunov exponent and apparently desynchronization tendency. In the performed set of numerical simulations, the synchronization measures were also evaluated for a two-dimensional parameter space defined by the neuronal coupling (emitter to a receiver neuron) and the (receiver) excitation current. Such analysis is repeated for different feedback couplings as well for different (emitter) excitation currents, revealing interesting characteristics of the attained synchronization region and conditions that facilitate the emergence of the synchronous behavior. These results provide a more detailed numerical insight of the underlying behavior of a HR in the excitation and coupling space, being in accordance with some general findings concerning HR coupling topologies. As a perspective, besides the synchronization overview from different standpoints, we hope that the proposed numerical approach for conditional Lyapunov exponent evaluation could outline a valuable strategy for studying neuronal stability, especially when realistic models are considered, in which analytical or even Jacobian evaluation could define a laborious or impracticable task.

Introduction

Since its formal discovery by Christiaan Huygens in 1665, the synchronization phenomenon has become one of the most important universal concepts in self-organization and emergence in nature [1]. Essentially, synchronization can be understood as the capacity of elements (or systems) to form a common temporal regime when weakly coupled, something that has been extensively observed and studied in different scientific areas (for an instigating overview, see [2]).

In the context of neuroscience, in particular, synchronization plays a key role in information processing, being associated with the emergence of coherent behavior and cognition [3], [4], [5]. For instance, if on the one hand the electrical synchronization (or desynchronization) of specific neuronal cells in the motor cortex can be related to particular movements (like those performed by the right or left hand), as classically considered for designing brain-computer interfaces [6], on the other hand, synchronization can be associated with pathological rhythms, such as those observed under epileptic seizures (see [7] for a deeper description of the synchronization and desynchronization role in epileptiform activity). Thus, it is not difficult to realize that the development of analytical and numerical tools to evaluate the synchronization between nonlinear oscillators in a quantitative way corresponds to an extremely important issue not just for neuroscience, but also for the study and analysis of nonlinear phenomena in general [1].

Despite being known since Huygens’ work, the synchronization of nonlinear oscillators and its stability, specially of those capable of generating chaotic behavior, has been suitably characterized only recently (in the1980′s and the1990′s) by the works of Yamada and Fujisaka [8] and Pecora and Carroll [9], with the introduction of the conditional Lyapunov exponents. Since then, the synchronization of nonlinear circuits [10], communication systems [11], mechanical systems and information transfer in neuronal systems [12], [13] has been extensively studied in such quantitative framework. Besides that, synchronism between nonlinear systems has also been studied by means of different perspectives, which includes measures to access nonlinear interdependence, spectral coherence, statistical similarities (e.g. cross-correlation) and higher order information theoretical measures (e.g. mutual information), phase coherence, spike distances, and, more recently, causal and information transfer measures (e.g. Granger causality, transfer entropy, partial directed coherence) [14], [15], [16], [17].

Having these issues in mind, this work aims, in the first place, to present a practical approach for conditional Lyapunov exponents calculation - preliminarly discussed in [18], [19] - in order to quantitatively characterize synchronization stability. To accomplish this task, the method of cloned dynamics (ClDyn) [20] to evaluate the classical Lyapunov spectrum - based on perturbation theory - is adapted for the synchronization analysis in the context of neuronal coupled oscillators. It is shown here that the ClDyn approach, which does not require the linearization of the dynamical system under study and the resolution of the variational equations underlying it, defines a convenient choice for conditional Lyapunov exponent estimation, specially when models with a complex mathematical description are considered [21], [22], [23]. In order to illustrate the proposed approach, we have evaluated the conditional Lyapunov exponent in the context of Duffing master-slave scenario, in which an analytical reference value can be obtained, as shown here.

After that, this alternative procedure for evaluating synchronism stability is employed to analyze the synchronism of chaotic bursting generated by Hindmarsh-Rose dynamical models with uni- and bi-directional coupling and with different excitation patterns, which included also included classical root mean square error (RMSE), correlation, mutual information and transfer entropy between observed neuronal activity. These analyses provide an insight on how the excitation and the coupling affect the synchronism and how a control parameter can be changed in order to match the common oscillatory behavior. In particular, it is shown here that increasing coupling in a weak condition does not necessarily increase the synchronous behavior, which was also shown by [24], although the transfer entropy can detect the neuronal emitter influence even under weak coupling condition and asynchronous state. In addition to that, it is also shown that synchronization is favored when the excitation of the receiving neuron is lower than the emitter, been such condition strengthen when a feedback coupling (from the receiving to the emitter) is introduced, which is in agreement with the main findings provided by [25] in a symmetrized network of HR neurons. It can be noted here in more detail that the synchronization region is shifted according to the excitation degree of the emitter, which can suggest a particular matching condition in order to favor transfer information between neurons.

This work is organized as follows: Section 2 presents a brief review of the neuronal model and the simulation scenarios analyzed here. Sections 3 and 4 present the cloned dynamics approach in the context of the estimation of the conditional Lyapunov exponent. Section 5 presents the statistical measures and the transfer entropy approach for studying synchronism from membrane potential observations. Section 6 brings the results concerning the stability analysis of the synchronization for the Hindmarsh-Rose in context of uni- and bi-directional coupling with asymmetric excitation. Finally, Section 7 presents the discussions and conclusions of the work.

Section snippets

The bursting phenomena and the Hindmarsh-Rose neuronal model

The Hindmarsh-Rose (HR) neuronal model was introduced by J. L. Hindmarsh and R. M. Rose in 1984 [26], based on modifications of the classical FitzHugh-Nagumo equations [27], [28], aiming to capture the bursting behavior, i.e. a firing pattern defined by periodic supra-threshold activity interleaved by electrical silence, both of irregular duration [29]. According to the original work of Hindmarsh and Rose, such bursting behavior finds strong experimental support in phenomena ranging from the

The cloned dynamics (cldyn) approach for Lyapunov exponent estimation

Lyapunov exponents are classically defined as the mean exponential divergence (or convergence) rate of initially close trajectories. The set of all Lyapunov exponents (also called Lyapunov spectrum) characterizes the topology of the flux for continuous time dynamical systems (i.e. if there is convergence to a fixed point, a limit cycle, a torus or even to a strange attractor) and corresponds to an invariant measure of the dynamics, being thus robust to the initial conditions or a specific

The cldyn approach in the context of synchronization analysis

Before applying the ClDyn approach in order to evaluate the synchronization stability, it is import to perform a change of coordinates aiming to capture the evolution of the synchronization error (as described in detail in [40]). For instance, consider the HR model presented in Eq. (1) coupled with an identical system by means of a resistive bidirectional connection (gap junction) as illustrated in Fig. 1. In this case, the energy transfer between the neuronal oscillators is defined by a

Statistical measures and transfer entropy

Besides synchronization stability measures, it is also valuable to introduce statistical measures that allow to capture synchronization from data. As complete synchronization conditions imply in perfect locking between systems, the root mean square error RMSE between membrane potentials (x1[n] and x2[n]) defines a natural option: RMSE=1Nn=1N[x1(n)x2(n)]2 in which N is the number of samples and n is the time index. Despite quantifying proximity, the RMSE offers absolute values, which can make

Results

In the following, firstly, an analytical value for conditional Lyapunov exponent is obtained for the Duffing model under master-slave operation [10] using the ClDyn approach. After that, representative set of simulations are analyzed in the context of HR neurons, which includes the uni- and bi-directional coupling under asymmetric excitation and synchronization analysis by means of conditional Lyapunov exponents, root mean square error, Pearson correlation, mutual information and transfer

Discussions and conclusions

A better understanding of the synchronization conditions and maximum information transfer in neuronal system is an issue of major relevance in neuroscience [15], [16]. In particular, synchronization stability has been extensively addressed in nonlinear theory, while statistical and information theoretical measures are more commonly used in the experimental neuroscience. Here we have presented a more detailed picture of the synchronization phenomenon by means of an alternative technique for

Acknowledgment

The authors thank CAPES and CNPq (grant n. 305621/2015-7, 449699/2014-5, 449467/2014-7, 305616/2016-1, 310610/2015-0) and FAPESP (grant n. 2013/07559-3)for the financial support.

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