Mathematical Problems in Engineering

Volume 2017, Article ID 7587294, 10 pages

https://doi.org/10.1155/2017/7587294

## Robust Master-Slave Synchronization of Neuronal Systems

^{1}División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana Azcapotzalco, Ciudad de México, Mexico^{2}Facultad de Ciencias Químicas, Universidad Veracruzana, Campus Xalapa, Xalapa, VER, Mexico^{3}Cátedras CONACyT, Universidad Autónoma Metropolitana Azcapotzalco, Ciudad de México, Mexico

Correspondence should be addressed to Hector Puebla; xm.mau.cza.oerroc@albeuph

Received 27 August 2017; Revised 4 December 2017; Accepted 11 December 2017; Published 28 December 2017

Academic Editor: Miguel A. F. Sanjuan

Copyright © 2017 Hector Puebla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Understanding how the brain works from a quantitative viewpoint is the domain of neural engineers [1]. Neural engineers apply mathematical and computational models, electrical engineering, and signal processing of living neuronal tissues [1, 2]. Two fundamental issues in neurosciences are the synchronization of individual neurons and the functional role of synchronized activity [3, 4]. Synchronization of neuron’s activities is necessary for memory, calculation, motion control, and diseases such as epilepsy [5–8].

Synchronized activity and temporal correlation are critical for encoding and exchanging information for neuronal information processing in the brain [2–4]. Synchronization approaches in neuronal systems are aimed at exploring the communication between neurons with the computing of coupling functions that resemble observed experimental electrical cell activity [6–10]. From the general synchronization point of view, synchronization approaches can be classified into two general groups [11, 12]: (i) natural coupling (self-synchronization) [13–21] and (ii) artificial coupling using state observers or feedback control approaches [22–34].

Classical approaches to the problem of neuronal synchronization include diffusive and phase couplings [12–21]. Diffusive coupling via gap junctions is considered as the natural form of coupling in many neuronal processes [19–21]. Gap junctions can be written as a particular form of diffusive coupling. Phase coupling consists of modeling each member of the population as a phase oscillator and coupling them through the sine of their phase differences [21]. For instance, Wang et al. [19] applied phase differences to study different states of synchrony in two electrically coupled neurons.

From control engineering, two ways for synchronization of nonlinear systems, including the case of neuronal systems, are (i) observer-based synchronization [11, 12, 22], which uses state observers to synchronize nonlinear oscillators with the same order and structure, reaching identical synchronization, and (ii) controller-based synchronization [23–34], which uses control laws to achieve the synchronization between nonlinear oscillators, with different structure and order.

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 [4, 35]. Moreover, experimental observations have pointed out that the synchronization phenomena in neuronal systems have robustness properties against cellular variability and intrinsic noise [36–40].

Relevant contributions addressing the synchronization of neuronal systems are the following. Aguilar-López and Martínez-Guerra [24] proposed a high order sliding mode controller that shows good robustness capabilities to external perturbations and internal noise. Bin et al. [25] introduced a backstepping control approach based on a Lyapunov function that achieves synchronization despite external disturbances. Based on feedback linearization ideas, Cornejo-Pérez and Femat [26] and Wang et al. [27, 28] introduced nonlinear controllers that achieve synchronization of coupled neurons despite external disturbances and unmeasured states. Nguyen and Hong [29] designed nonlinear and linear controllers with parameter adaptation to consider parameter uncertainties. They achieved synchronization of two coupled neurons. Using MPC and optimal controllers, Fröhlich and Jezernik [30, 31] designed controllers for the suppression of oscillations in neurons. Rehan et al. [23] and Rehan and Hong [32] proposed robust synchronization approaches using a linear matrix inequality controller and adaptation laws for uncertain parameters. Puebla et al. [33] introduced a robust feedback control scheme endowed with uncertainty compensation for regulation and tracking tasks in coupled neurons. Wang and Zhao [34] proposed a nonlinear model-based controller based on the inversion of the dynamics which guarantees the synchronization under no parametric uncertainties. Most of the above papers have addressed the robust synchronization problem of neuronal systems. However, their practical application is limited because of their structure and high computing cost as well as involved control designs.

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 [41]. There are different types of synchronization for coupled systems [11–13]. In this paper, synchronization of neuronal systems is defined as the match of the membrane potential. It is found that the MEC approach may achieve robust synchronization of the membrane potential via a coupling function also applied to the membrane potential. Numerical simulations on two benchmark neuronal systems show good performance of the robust synchronizer design.

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 [41, 42], including the effect of interconnection dynamics induced by model uncertainties. (iv) Our approach has a simple structure and provides good robustness against external perturbations and noise facilitating its physiological interpretation.

The rest of this work is organized as follows. In Section 2, we introduce two benchmark case studies of neuronal systems. In Section 3, the proposed robust master-slave synchronization is introduced. Section 4 presents the implementation and performance of the robust master-slave synchronization approach. Finally, in Section 5, we provide some concluding remarks.

#### 2. Modeling Neurons

Mathematical modeling has made an enormous impact on neuroscience [1–4, 35]. A variety of dynamic models of the electrical activity of neurons have been reported in the literature [2–4, 43–46]. In this section, we introduce two benchmark case studies of neuronal systems: (i) the model proposed by Hodgkin and Huxley (HH) [35, 43–46], which is a realistic neuron model describing the propagation of an electric pulse along a squid axon membrane, and (ii) the Hindmarsh-Rose (HR) neuron model based on Hodgkin-Huxley type models describing the signal transmission across axons in neurobiology [2, 16, 35]. Based on the model structure of case studies, a general model of coupled or uncoupled neuronal systems is introduced. For completeness, we provide a brief introduction to the modeling of neurons.

##### 2.1. Modeling the Electrical Activity of Neurons

The nervous system of an organism, which consists of neurons, is a communication network that allows for rapid transmission of information between cells [2, 16, 35]. A neuron receives information through the dendrites which are transported via axons, which provide links to other neurons via synapses [2, 16]. The transport of ions of sodium and potassium through the outer membrane of a nerve cell is responsible for electrical signals that transmit information to other neurons [2, 16].

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 [2, 16, 35].

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 [43–45]. In particular, sodium and potassium ions are fundamental in the functioning of a neuron [2, 16, 35]. The cell membrane of a neuron is impermeable to sodium and potassium ions when the cell is in a resting state. An inactive neuron has a resting potential, which is generated via a transport protein called the sodium-potassium pump. This protein moves sodium ions outside the cell and simultaneously moves some potassium ions into the cell’s cytoplasm. Thus, the cell is more positive outside than inside, due to the fact that the number of sodium ions moved outside the cell is greater than the number of potassium ions moved inside. When a stimulus arrives at the nerve cell, its surface becomes permeable to sodium ions, which flows into the cells, resulting in a reversal of polarization. The interior of the cell becomes positively charged, and the outside becomes negatively charged. Then, the interior becomes permeable to potassium which flows outside through potassium channels, reversing the polarization of the cell below the polarization of the resting state. To restore that polarization, the excess of the cell sodium (at the interior) and potassium (at the exterior) is pumped [2, 16, 35, 43–46].

HH described the action potential wave of excited squid giant axons with an external electrical signal via a set of mathematical equations [2, 16, 35, 43–46]. At present, it is still the basic model for describing such phenomena [2, 16, 35]. The HH model for excitability in the membrane of the squid giant axon is complicated and consists of one nonlinear partial differential equation coupled to three ordinary differential equations [43–45].

In the early 1960s, FitzHugh applied model reduction techniques to the analysis of the HH equations [45]. That reduction of the HH equations later became known as the FitzHugh-Nagumo (FHN) model and had given a great insight into the mathematical and physiological complexities of the excitability process [2, 16, 45]. The FHN model reduction uses the fact that the time scales of the two channels are quite different. Sodium channel had a faster time scale than potassium channel. Thus, the sodium channel can always be considered in equilibrium, reducing the HH model to two equations [45]. Thus, FHN model is an approximation to the HH model retaining essential features of the action potential.

##### 2.2. HH Neuron Model

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 [44]. The HH model for two neurons is described by the following set of eight ordinary differential equations (ODEs) [2, 16, 35]: where , , , and (subindices denote master and slave neurons) represent the membrane potential, the activation of the potassium flow current, and the activation and inactivation of the sodium flow current, respectively. is the membrane capacitance, , , and are the maximum ionic and leak conductance, and , , and stand for the ionic and leak reversal potentials [2, 4, 16, 39]. is the external stimulus current. The functions and describe the transition rates between open and closed states of the channels.

##### 2.3. Hindmarsh-Rose Neurons

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 [2, 16, 35]. Under external current stimulation, the individual HR model may show chaotic behavior. The model of two uncoupled HR neurons is described as where is the membrane potential, is associated with the fast current Na^{+} or K^{+}, and is associated with the slow current, for example, Ca^{2+}. is the external current input.

##### 2.4. A General Model of Synchronized Neuronal Systems

We consider a general class of master-slave configuration of neuronal systems coupled through the membrane potential, that is, . The dynamics of the master neuron are modeled aswhere denotes the membrane potential of the master neuron and are the remaining states of the master neuron.

The dynamics of the slave neuron are modeled as where denotes the membrane potential of the slave neuron and are the remaining states of the slave neuron.

Coupled neurons can be modeled aswhere denotes the synchronization error and is an external electrical input applied to the slave neuron.

The following comments are in order:(i)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 [2, 16, 35, 43–46]. Both HH and HR neuron models can reproduce its main features when they are exposed to an external current (existence of an excitation threshold, relative and absolute refractory periods, and the generation of pulse trains). Thus, for synchronization design purposes, benchmark models with small dimension and less complexity are more suitable.(ii)The external input represents an externally applied current into the cell from an electrode. The membrane voltage can also be readily measured, and the controller can be realized easily using this combination of input-output variables. The use of an external current as the manipulable variable is realistic since it has a significant effect on the dynamics of membrane potential leading to depolarization and repolarization of the neuron [2, 4]. On the other hand, several experimental studies have shown that the synchronization of coupled neurons depends on external stimulus properties [10, 13–15].(iii)Uncertainties in neuron models arise in two main ways: structural and parametric [35, 40, 43]. Structural uncertainty refers to different choices of fitting of sodium and potassium conductance curves in a model. The uncertainty that arises from the approximation of complex models to simpler ones also fits into the category of model uncertainty. Parametric uncertainty refers to variation in the numerical base values of different parameters of the model. These parameters may include changes due to intrinsic electric and magnetic properties of tissue. For instance, each neuron may have a different set of conductances [16, 35, 40]. Moreover, the thermal motion of the molecules leads to noise and fluctuations in the variables of the model [36–39].

#### 3. Robust Master-Slave Synchronization

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.

##### 3.1. Synchronization Problem

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 1 shows the scheme of the synchronization approach.