Analysis of Chaotic Resonance in Izhikevich Neuron Model

Master stability functions of networks of Izhikevich neurons

Synchronization has attracted interest in many areas where the systems under study can be described by complex networks. Among such areas is neuroscience, where it is hypothesized that synchronization plays a role in many functions and dysfunctions of the brain. We study the linear stability of synchronized states in networks of Izhikevich neurons using master stability functions (MSFs), and to accomplish that, we exploit the formalism of saltation matrices. Such a tool allows us to calculate the Lyapunov exponents of the MSF properly since the Izhikevich model displays a discontinuity within its spikes. We consider both electrical and chemical couplings as well as global and cluster synchronized states. The MSF calculations are compared with a measure of the synchronization error for simulated networks. We give special attention to the case of electric and chemical coupling, where a riddled basin of attraction makes the synchronized solution more sensitive to perturbations.

Parameter and coupling estimation in small networks of Izhikevich's neurons

Nowadays, experimental techniques allow scientists to have access to large amounts of data. In order to obtain reliable information from the complex systems that produce these data, appropriate analysis tools are needed. The Kalman filter is a frequently used technique to infer, assuming a model of the system, the parameters of the model from uncertain observations. A well-known implementation of the Kalman filter, the unscented Kalman filter (UKF), was recently shown to be able to infer the connectivity of a set of coupled chaotic oscillators. In this work, we test whether the UKF can also reconstruct the connectivity of small groups of coupled neurons when their links are either electrical or chemical synapses. In particular, we consider Izhikevich neurons and aim to infer which neurons influence each other, considering simulated spike trains as the experimental observations used by the UKF. First, we verify that the UKF can recover the parameters of a single neuron, even when the parameters vary in time. Second, we analyze small neural ensembles and demonstrate that the UKF allows inferring the connectivity between the neurons, even for heterogeneous, directed, and temporally evolving networks. Our results show that time-dependent parameter and coupling estimation is possible in this nonlinearly coupled system.

Chaotic and stochastic dynamics of epileptiform-like activities in sclerotic hippocampus resected from patients with pharmacoresistant epilepsy

The types of epileptiform activity occurring in the sclerotic hippocampus with highest incidence are interictal-like events (II) and periodic ictal spiking (PIS). These activities are classified according to their event rates, but it is still unclear if these rate differences are consequences of underlying physiological mechanisms. Identifying new and more specific information related to these two activities may bring insights to a better understanding about the epileptogenic process and new diagnosis. We applied Poincaré map analysis and Recurrence Quantification Analysis (RQA) onto 35 in vitro electrophysiological signals recorded from slices of 12 hippocampal tissues surgically resected from patients with pharmacoresistant temporal lobe epilepsy. These analyzes showed that the II activity is related to chaotic dynamics, whereas the PIS activity is related to deterministic periodic dynamics. Additionally, it indicates that their different rates are consequence of different endogenous dynamics. Finally, by using two computational models we were able to simulate the transition between II and PIS activities. The RQA was applied to different periods of these simulations to compare the recurrences between artificial and real signals, showing that different ranges of regularity-chaoticity can be directly associated with the generation of PIS and II activities.

Temporal lobe epilepsy (TLE) is the most prevalent type of epilepsy in adults and hippocampal sclerosis is the major pathophysiological substrate of pharmaco-refractory TLE. Different patterns of epileptiform-like activity have been described in human hippocampal sclerosis, but the standard analysis applied to characterize the activities usually do not consider the nonlinear features that epileptiform patterns exhibit. Here, using Poincaré map and Recurrence Quantitative Analysis we characterized the most prevalent type of epileptiform-like activities—interictal-like events (II) and periodic ictal spiking (PIS), recorded in vitro from resected hippocampi of pharmacoresistant patients with TLE—according to their levels of stochasticity, chaoticity and determinism. The II activities showed to be more chaotic with complex rhythmicity than PIS activities. The nonlinear dynamic differences between II and PIS leads us to conjecture that they are expressions of different seizure susceptibility. We also identified that each hippocampal subfield expresses II and PIS activities in a specific and different way. Finally, from the modulation of internal parameters of two computational models, we show the conversion of one type of activity into the other, showing how specific neuron networks synchronize over time, leading to II and PIS activities and then into a generalized seizure.

Effects of chaotic activity and time delay on signal transmission in FitzHugh-Nagumo neuronal system

The influences of chaotic activity and time delay on the transmission of the sub-threshold signal (STS) in a single FitzHugh-Nagumo neuron and coupled neuronal networks are studied. It is found that a moderate chaotic activity level can enhance the system's detection and transmission of STS. This phenomenon is known as chaotic resonance (CR). In a single neuron, the large amplitude and small period of the STS have a positive effect on the CR phenomenon. In the coupled neuronal network, however, the signal transmission performance of chemical synapses is better than that of electrical synapses. The time delay can determine the trend of the system response, and the multiple chaotic resonances phenomenon is observed upon fine-tuning the time delay length. Both sub-harmonic chaotic resonance and chaotic anti-resonance appear when the STS period and time delay are locked. In chained networks, the signal transmission performance between electrical synapses attenuates continuously. Conversely, the performance between chemical synapses reaches a steady state.

Vibrational resonance in a neuron-astrocyte coupled model

Recent findings have revealed that not only neurons but also astrocytes, a special type of glial cells, are major players of neuronal information processing. It is now widely accepted that they contribute to the regulation of their microenvironment by cross-talking with neurons via gliotransmitters. In this context, we here study the phenomenon of vibrational resonance in neurons by considering their interaction with astrocytes. Our analysis of a neuron-astrocyte pair reveals that intracellular dynamics of astrocytes can induce a double vibrational resonance effect in the weak signal detection performance of a neuron, exhibiting two distinct wells centred at different high-frequency driving amplitudes. We also identify the underlying mechanism of this behaviour, showing that the interaction of widely separated time scales of neurons, astrocytes and driving signals is the key factor for the emergence and control of double vibrational resonance. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.

Noise suppression ability and its mechanism analysis of scale-free spiking neural network under white Gaussian noise

With the continuous improvement of automation and informatization, the electromagnetic environment has become increasingly complex. Traditional protection methods for electronic systems are facing with serious challenges. Biological nervous system has the self-adaptive advantages under the regulation of the nervous system. It is necessary to explore a new thought on electromagnetic protection by drawing from the self-adaptive advantage of the biological nervous system. In this study, the scale-free spiking neural network (SFSNN) is constructed, in which the Izhikevich neuron model is employed as a node, and the synaptic plasticity model including excitatory and inhibitory synapses is employed as an edge. Under white Gaussian noise, the noise suppression abilities of the SFSNNs with the high average clustering coefficient (ACC) and the SFSNNs with the low ACC are studied comparatively. The noise suppression mechanism of the SFSNN is explored. The experiment results demonstrate that the following. (1) The SFSNN has a certain degree of noise suppression ability, and the SFSNNs with the high ACC have higher noise suppression performance than the SFSNNs with the low ACC. (2) The neural information processing of the SFSNN is the linkage effect of dynamic changes in neuron firing, synaptic weight and topological characteristics. (3) The synaptic plasticity is the intrinsic factor of the noise suppression ability of the SFSNN.

Multiple Firing Patterns in Coupled Hindmarsh-Rose Neurons with a Nonsmooth Memristor

A model is introduced by coupling two three-dimensional Hindmarsh-Rose models with the help of a nonsmooth memristor. The firing patterns dependent on the external forcing current are explored, which undergo a process from adding-period to chaos. The stability of equilibrium points of the considered model is investigated via qualitative analysis, from which it can be gained that the model has diversity in the number and stability of equilibrium points for different coupling coefficients. The coexistence of multiple firing patterns relative to initial values is revealed, which means that the referred model can appear various firing patterns with the change of the initial value. Multiple firing patterns of the addressed neuron model induced by different scales are uncovered, which suggests that the discussed model has a multiscale effect for the nonzero initial value.

Resonance phenomena controlled by external feedback signals and additive noise in neural systems

Chaotic resonance is a phenomenon that can replace the fluctuation source in stochastic resonance from additive noise to chaos. We previously developed a method to control the chaotic state for suitably generating chaotic resonance by external feedback even when the external adjustment of chaos is difficult, establishing a method named reduced region of orbit (RRO) feedback. However, a feedback signal was utilized only for dividing the merged attractor. In addition, the signal sensitivity in chaotic resonance induced by feedback signals and that of stochastic resonance by additive noise have not been compared. To merge the separated attractor, we propose a negative strength of the RRO feedback signal in a discrete neural system which is composed of excitatory and inhibitory neurons. We evaluate the features of chaotic resonance and compare it to stochastic resonance. The RRO feedback signal with negative strength can merge the separated attractor and induce chaotic resonance. We also confirm that additive noise induces stochastic resonance through attractor merging. The comparison of these resonance modalities verifies that chaotic resonance provides more applicability than stochastic resonance given its capability to handle attractor separation and merging.

Formation of Opioid-Induced Memory and Its Prevention: A Computational Study

There are several experimental studies which suggest opioids consumption forms pathological memories in different brain regions. For example it has been empirically demonstrated that the theta rhythm which appears during chronic opioid consumption is correlated with the addiction memory formation. In this paper, we present a minimal computational model that shows how opioids can change firing patterns of the neurons during acute and chronic opioid consumption and also during withdrawal periods. The model consists of a pre- and post-synaptic neuronal circuits and the astrocyte that monitors the synapses. The output circuitry consists of inhibitory interneurons and excitatory pyramidal neurons. Our simulation results demonstrate that acute opioid consumption induces synchronous patterns in the beta frequency range, while, chronic opioid consumption provokes theta frequency oscillations. This allows us to infer that the theta rhythm appeared during chronic treatment can be an indication of brain engagement in opioid-induced memory formation. Our results also suggest that changing the inputs of the interneurons and the inhibitory neuronal network is not an appropriate method for preventing the formation of pathological memory. However, the same results suggest that prevention of pathological memory formation is possible by manipulating the input of the stimulatory network and the excitatory connections in the neuronal network. They also show that during withdrawal periods, firing rate is reduced and random fluctuations are generated in the modeled neural network. The random fluctuations disappear and synchronized patterns emerge when the activities of the astrocytic transporters are decreased. These results suggest that formation of the synchronized activities can be correlated with the relapse. Our model also predicts that reduction in gliotransmitter release can eliminate the synchrony and thereby it can reduce the likelihood of the relapse occurrence.

Routes to Chaos Induced by a Discontinuous Resetting Process in a Hybrid Spiking Neuron Model

Several hybrid spiking neuron models combining continuous spike generation mechanisms and discontinuous resetting processes following spiking have been proposed. The Izhikevich neuron model, for example, can reproduce many spiking patterns. This model clearly possesses various types of bifurcations and routes to chaos under the effect of a state-dependent jump in the resetting process. In this study, we focus further on the relation between chaotic behaviour and the state-dependent jump, approaching the subject by comparing spiking neuron model versions with and without the resetting process. We first adopt a continuous two-dimensional spiking neuron model in which the orbit in the spiking state does not exhibit divergent behaviour. We then insert the resetting process into the model. An evaluation using the Lyapunov exponent with a saltation matrix and a characteristic multiplier of the Poincar'e map reveals that two types of chaotic behaviour (i.e. bursting chaotic spikes and near-period-two chaotic spikes) are induced by the resetting process. In addition, we confirm that this chaotic bursting state is generated from the periodic spiking state because of the slow- and fast-scale dynamics that arise when jumping to the hyperpolarization and depolarization regions, respectively.

Chaotic Resonance in Typical Routes to Chaos in the Izhikevich Neuron Model

Chaotic resonance (CR), in which a system responds to a weak signal through the effects of chaotic activities, is a known function of chaos in neural systems. The current belief suggests that chaotic states are induced by different routes to chaos in spiking neural systems. However, few studies have compared the efficiency of signal responses in CR across the different chaotic states in spiking neural systems. We focused herein on the Izhikevich neuron model, comparing the characteristics of CR in the chaotic states arising through the period-doubling or tangent bifurcation routes. We found that the signal response in CR had a unimodal maximum with respect to the stability of chaotic orbits in the tested chaotic states. Furthermore, the efficiency of signal responses at the edge of chaos became especially high as a result of synchronization between the input signal and the periodic component in chaotic spiking activity.

Chaotic Resonance in Coupled Inferior Olive Neurons with the Llinás Approach Neuron Model

It is well known that cerebellar motor control is fine-tuned by the learning process adjusted according to rich error signals from inferior olive (IO) neurons. Schweighofer and colleagues proposed that these signals can be produced by chaotic irregular firing in the IO neuron assembly; such chaotic resonance (CR) was replicated in their computer demonstration of a Hodgkin-Huxley (HH)-type compartment model. In this study, we examined the response of CR to a periodic signal in the IO neuron assembly comprising the Llinás approach IO neuron model. This system involves empirically observed dynamics of the IO membrane potential and is simpler than the HH-type compartment model. We then clarified its dependence on electrical coupling strength, input signal strength, and frequency. Furthermore, we compared the physiological validity for IO neurons such as low firing rate and sustaining subthreshold oscillation between CR and conventional stochastic resonance (SR) and examined the consistency with asynchronous firings indicated by the previous model-based studies in the cerebellar learning process. In addition, the signal response of CR and SR was investigated in a large neuron assembly. As the result, we confirmed that CR was consistent with the above IO neuron's characteristics, but it was not as easy for SR.