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The desire to understand physiological mechanisms of neuronal systems has led to the introduction of engineering concepts to explain how the brain works. The synchronization of neurons is a central topic in understanding the behavior of living organisms in neurosciences and has been addressed using concepts from control engineering. We introduce a simple and reliable robust synchronization approach for neuronal systems. The proposed synchronization method is based on a master-slave configuration in conjunction with a coupling input enhanced with compensation of model uncertainties. Our approach has two nice features for the synchronization of neuronal systems: (i) a simple structure that uses the minimum information and (ii) good robustness properties against model uncertainties and noise. Two benchmark neuronal systems, Hodgkin-Huxley and Hindmarsh-Rose neurons, are used to illustrate our findings. The proposed synchronization approach is aimed at gaining insight into the effect of external electrical stimulation of nerve cells.

Understanding how the brain works from a quantitative viewpoint is the domain of neural engineers [

Synchronized activity and temporal correlation are critical for encoding and exchanging information for neuronal information processing in the brain [

Classical approaches to the problem of neuronal synchronization include diffusive and phase couplings [

From control engineering, two ways for synchronization of nonlinear systems, including the case of neuronal systems, are (i) observer-based synchronization [

Control designs pose significant challenges due to the presence of disturbances, dynamic uncertainties, and nonlinearities in neuronal models. Indeed, neuronal models have significant structural and parametric uncertainties. For instance, cell capacitances and resistances are obtained from biophysical data obtained from diverse sources [

Relevant contributions addressing the synchronization of neuronal systems are the following. Aguilar-López and Martínez-Guerra [

A particular configuration for controller designs is the master-slave synchronization configuration, where the variable states of slave neurons are forced to follow the trajectories of a master neuron, which leads to an autonomous synchronization error. In this work, we address the master-slave synchronization of neuronal systems using a robust approach based on modeling error compensation (MEC) ideas [

The main contributions of this work can be summarized in four aspects. (i) We derive our control approach based on the direct dynamics of the master-slave synchronization error, leading to an autonomous tracking error and avoiding the change of coordinates as in feedback linearization and backstepping approaches. (ii) The proposed robust synchronization approach uses the minimum systems information (only the membrane potential measurement), and the coupling signal is also injected only to the membrane potential, facilitating its implementation in real systems. (iii) We use singular-perturbation theory as our main nonlinear stability tool [

The rest of this work is organized as follows. In Section

Mathematical modeling has made an enormous impact on neuroscience [

The nervous system of an organism, which consists of neurons, is a communication network that allows for rapid transmission of information between cells [

Neurons are excitable media and respond to electrical stimuli, and this response is exploited when studying neurons. After a low impact of electric current, the excitable cells relax immediately to their initial state. If a pulse exceeds a threshold value, a single nerve pulse appears on the excitable membrane of the nerve tissue (action potential) that propagates along the nerve, preserving constant amplitude and form [

The propagation mechanism of an electric pulse along a membrane axon is associated with the fact that the permittivity of a membrane depends on existing currents and voltages and is different for different ions [

HH described the action potential wave of excited squid giant axons with an external electrical signal via a set of mathematical equations [

In the early 1960s, FitzHugh applied model reduction techniques to the analysis of the HH equations [

The HH neurons are usually used as realistic models of neuronal systems, for studying neuronal synchronization. The HH model describes how action potentials in neurons are initiated and propagated and approximates the electrical characteristics of excitable cells [

As a second case study, we consider a benchmark Hindmarsh-Rose (HR) neuron model, which can be seen as a physiologically realistic model of the HH type describing the signal transmission across axons in neurobiology [^{+} or K^{+}, and ^{2+}.

We consider a general class of master-slave configuration of neuronal systems coupled through the membrane potential, that is,

The dynamics of the slave neuron are modeled as

Coupled neurons can be modeled as

The following comments are in order:

The original HH model is given by coupled nonlinear ODEs which are a simplification of full partial differential equations (PDEs) that describes the neuron membrane [

The external input

Uncertainties in neuron models arise in two main ways: structural and parametric [

In this section, based on modeling error compensation (MEC) ideas, the synchronizer design is presented. First, the problem is stated as a master-slave configuration, and some assumptions for the synchronizer design are introduced. Next, robustness and stability issues of the synchronization approach are provided.

The synchronization problem is stated as follows; that is, the output of a master neuron is the reference of a slave neuron so that the output of the slave system follows the output of the master system asymptotically. We apply an external signal at the slave neuron to track the desired behavior of the master neuron. Figure

Master-slave synchronization of neuronal systems.

The following assumptions complete the synchronization problem description:

Nonlinear functions

The general coupled neuron model given by (

The measurement of the membrane voltage in the master and slave neurons is available for synchronization design purposes.

The following comments are in order:

(A1) is realistic. Indeed, the primary source of nonlinearity in neuron models is the conductance curves, which meet these assumptions [

(A2) considers that the coupled neuron model contains uncertainties related to uncertain parameters and unmodeled dynamics, that is,

The synchronizer design consists of the following steps.

Consider the coupled neurons model given by (

Lump the uncertain terms in a single new state

Estimate the uncertain term

Design a synchronizer to drive the synchronization error to zero with the dynamics given by

The resulting synchronizer depends only on the measures of the membrane voltages in the master and slave neurons and the estimated value of the lumped uncertain terms

The tuning of both parameters follows a simple rule [

To obtain satisfactory and practical synchronization strategies, they should be robust in response to both model uncertainties and external perturbations. The robustness properties against model uncertainties of the proposed synchronizer design are related to the compensation of the estimated lumped uncertain terms.

The stability analysis of the proposed synchronizer design is based on singular-perturbation arguments [

Given the synchronization error

In this section, simulation results are presented for the synchronization of the case studies. First, the proposed synchronizer approach is presented for three sets of synchronizer parameters

We consider two HH neurons with the following form of the functions ^{2}, ^{2}, and ^{2}, ^{2} (representing the maximum conductance of the corresponding ionic currents), ^{2} (membrane capacitance),

Figure

Synchronization of HH neurons for three sets of synchronizer parameters.

As shown in Figure

From Figure

The robustness capabilities of the MEC synchronization scheme against parameter mismatch and fluctuations in the membrane potential are evaluated as follows. (i) A random parameter mismatch of 5% between master and slave neurons is first considered. (ii) A random fluctuation of 10% was added in the membrane potential of the slave neuron. The above perturbations are simulated with Gaussian random noise, which is usually used to simulate most common disturbances in neuroscience [

The simulation results are shown in Figures

Robustness of the synchronization of HH neurons against parameter mismatch.

Robustness of the synchronization of HH neurons against noise.

For the second case study, we consider two HR neurons. Base parameter values are [

Figure

Synchronization of HR neurons for three sets of synchronizer parameters.

Figure

Figures

Robustness of the synchronization of HR neurons against parameter mismatch.

Robustness of the synchronization of HR neurons against noise.

The extension of the proposed synchronizer approach for three coupled neurons (i.e., two slave neurons) is illustrated in Figure

Synchronization of three HR neurons.

Figure

This paper introduces a robust approach for synchronization of neuronal systems. Using a master-slave configuration, we provide robustness capabilities via the lumping, estimation, and compensation of model uncertainties. The coupling function computing via the synchronization approach uses only the membrane potential and is only also applied to the membrane potential of the neuron, resembling the strength of electrical gap junctions. Synchronization dynamics are analyzed using stability arguments of nonsingular perturbation systems. The performance of the proposed synchronization approach is validated through in-depth numerical simulations on two benchmark models of neuronal systems. Furthermore, since our approach uses the minimum model information, the proposed method can be applied for synchronization of more complex and multiple neuronal systems. Our study aims to contribute to the understanding of both processes that influence the synchronization of individual neurons and the functional role of synchronized activity of coupled neurons in neural and mental disorders.

The authors declare that they have no conflicts of interest.