Coupled Model Neurons Research Articles

Averaging, Folded Singularities, and Torus Canards: Explaining Transitions between Bursting and Spiking in a Coupled Neuron Model

In this work, we identify generic bifurcation scenarios corresponding to transitions between bursting and tonic spiking solutions in a model for a coupled pair of burst-capable neurons, and we elucidate the central role of folded singularities in these scenarios. The folded singularities in our work arise in the context of fast-slow averaging, and hence our results link with the study of torus canards, a recently identified class of ordinary differential equation (ODE) solutions featuring oscillatory excursions along repelling structures in phase space [J. Burke et al., J. Math. Neurosci., 2 (2012), pp. 1--30]; in particular, our work extends this study to systems featuring two slow variables and symmetry and goes significantly beyond the analysis of activity transitions presented by Best et al. [SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 1107--1139].

Emergent patterns in interacting neuronal sub-populations

We investigate an ensemble of coupled model neurons, consisting of groups of varying sizes and intrinsic dynamics, ranging from periodic to chaotic, where the inter-group coupling interaction is effectively like a dynamic signal from a different sub-population. We observe that the minority group can significantly influence the majority group. For instance, when a small chaotic group is coupled to a large periodic group, the chaotic group de-synchronizes. However, counter-intuitively, when a small periodic group couples strongly to a large chaotic group, it leads to complete synchronization in the majority chaotic population, which also spikes at the frequency of the small periodic group. It then appears that the small group of periodic neurons can act like a pacemaker for the whole network. Further, we report the existence of varied clustering patterns, ranging from sets of synchronized clusters to anti-phase clusters, governed by the interplay of the relative sizes and dynamics of the sub-populations. So these results have relevance in understanding how a group can influence the synchrony of another group of dynamically different elements, reminiscent of event-related synchronization/de-synchronization in complex networks.

The interplay between STDP rules and anticipated synchronization in the organization of neuronal networks

Neuronal networks are able to synchronize in phase-locking regimes. Depending on the structural connectivity it is possible to determine the sign of the phase difference between oscillatory nodes. Specially in unidirectionally coupled motifs the relative phase difference is usually positive. However it has been shown that in the presence of dynamical inhibitory loops, unidirectionally coupled neuron models may present both positive or negative phase difference [1]. The unusual regime in which the phase difference is negative is called anticipated synchronization (AS) [2]. It means that two neurons or two neuronal populations coupled in a master-slave configuration can oscillate in such a way that the slave leads the master. These results are important in light of the growing experimental evidence that the synaptic strength between neurons can undergo spike-timing-dependent plasticity (STDP) [3]. In the delayed synchronization regime the master (pre-synaptic) neuron fires a spike before the slave (post-synaptic) neuron, which under STDP rules would facilitate long term potentiation (LTP), whereas in the AS regime the slave neuron fires a spike before the master neuron, contributing to long term depression (LTD) [4]. Since it has been shown previously that a simple 3-neuron motif can undergo a continuous transition from positive to negative values of relative phase via changes in synaptic conductances [1], the interplay between these regimes and STDP mechanisms is likely to play a very significant role in the organization of the system dynamics. Considering that in certain situations the relative potentiation in strong synapses is less intense than in weak synapses, but the depression do not show this dependence [5], we compare the effects of additive, multiplicative and hybrid rules in our motifs. Together with AS regimes in neuronal populations, STDP rules give synaptic weight distributions that are comparable to experiments in cortex [6,7].

Failure of Delayed Feedback Deep Brain Stimulation for Intermittent Pathological Synchronization in Parkinson’s Disease

Suppression of excessively synchronous beta-band oscillatory activity in the brain is believed to suppress hypokinetic motor symptoms of Parkinson’s disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize the synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson’s disease exhibits complex intermittent synchronous patterns, far from the idealized synchronous dynamics used to study the delayed feedback stimulation. This study explores the action of delayed feedback stimulation on partially synchronized oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a computational model of the basal ganglia networks which reproduces experimentally observed fine temporal structure of the synchronous dynamics. When the parameters of our model are such that the synchrony is unphysiologically strong, the feedback exerts a desynchronizing action. However, when the network is tuned to reproduce the highly variable temporal patterns observed experimentally, the same kind of delayed feedback may actually increase the synchrony. As network parameters are changed from the range which produces complete synchrony to those favoring less synchronous dynamics, desynchronizing delayed feedback may gradually turn into synchronizing stimulation. This suggests that delayed feedback DBS in Parkinson’s disease may boost rather than suppress synchronization and is unlikely to be clinically successful. The study also indicates that delayed feedback stimulation may not necessarily exhibit a desynchronization effect when acting on a physiologically realistic partially synchronous dynamics, and provides an example of how to estimate the stimulation effect.

Stable Antiphase Oscillations in a Network of Electrically Coupled Model Neurons

We consider a model of two identical neurons with electrical coupling and give a rigorous analysis for when the network exhibits stable antiphase behavior. Each neuron is modeled as a relaxation oscillator, and the main technique used in the analysis is geometric singular perturbation theory. Using fast/slow analysis, we derive a one-dimensional map; the antiphase solution corresponds to a fixed point of this map. Detailed analysis of the map, along with its derivative, leads to precise conditions on the duty cycle of an individual cell for when there exists an antiphase solution and when it is stable, for sufficiently weak electrical coupling. The results presented here extend our previous analysis [D. Terman et al., SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 1127--1153] to more biophysical neuronal models.

Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.

Qualitative validation of the reduction from two reciprocally coupled neurons to one self-coupled neuron in a respiratory network model

The pre-Bötzinger complex of the mammalian brainstem is a heterogeneous neuronal network, and individual neurons within the network have varying strengths of the persistent sodium and calcium-activated nonspecific cationic currents. Individually, these currents have been the focus of modeling efforts. Previously, Dunmyre et al. (J Comput Neurosci 1-24, 2011) proposed a model and studied the interactions of these currents within one self-coupled neuron. In this work, I consider two identical, reciprocally coupled model neurons and validate the reduction to the self-coupled case. I find that all of the dynamics of the two model neuron network and the regions of parameter space where these distinct dynamics are found are qualitatively preserved in the reduction to the self-coupled case.

Amplitude death in nonlinear oscillators with nonlinear coupling

Amplitude death is the cessation of oscillations that occurs in coupled nonlinear systems when fixed points are stabilized as a consequence of the interaction. We show here that this phenomenon is very general: it occurs in nonlinearly coupled systems in the absence of parameter mismatch or time delay although time-delayed interactions can enhance the effect. Application is made to synaptically coupled model neurons, nonlinearly coupled Rössler oscillators, as well as to networks of nonlinear oscillators with nonlinear coupling. By suitably designing the nonlinear coupling, arbitrary steady states can be stabilized.

A computational study of the interdependencies between neuronal impulse pattern, noise effects and synchronization

Alterations of individual neurons dynamics and associated changes of the activity pattern, especially the transition from tonic firing (single-spikes) to bursts discharges (impulse groups), play an important role for neuronal information processing and synchronization in many physiological processes (sensory encoding, information binding, hormone release, sleep–wake cycles) as well as in disease (Parkinson, epilepsy). We have used Hodgkin–Huxley-type model neurons with subthreshold oscillations to examine the impact of noise on neuronal encoding and thereby have seen significant differences depending on noise implementation as well as on the neuron’s dynamic state. The importance of the individual neurons’ dynamics is further elucidated by simulation studies with electrotonically coupled model neurons which revealed mutual interdependencies between the alterations of the network’s coupling strength and neurons’ activity patterns with regard to synchronization. Remarkably, a pacemaker-like activity pattern which revealed to be much more noise sensitive than the bursting patterns also requires much higher coupling strengths for synchronization. This seemingly simple pattern is obviously governed by more complex dynamics than expected from a conventional pacemaker which may explain why neurons more easily synchronize in the bursting than in the tonic firing mode.

The Design of Central Pattern Generators Based on the Matsuoka Oscillator to Generate Rhythmic Human-Like Movement for Biped Robots

We propose a controller based on a central pattern generator (CPG) network of mutually coupled Matsuoka nonlinear neural oscillators to generate rhythmic human-like movement for biped robots. The parameters of mutually inhibited and coupled Matsuoka oscillators and the necessary interconnection coupling coefficients within the CPG network directly influence the generation of the required rhythmic signals related to targeted motion. Our objective is to analyze the mutually coupled neuron models of Matsuoka oscillators to realize an efficient CPG design that leads to have dynamic, stable, sustained rhythmic movement with robust gaits for bipedal robots. We discuss the design of a CPG model with new interconnection coupling links and its inhibitation coefficients for a CPG-based controller. The new design was studied through interaction between simulated interconnection coupling dynamics with six links and a musculoskeletal model with the 6 degrees of freedom (DOFs) of a biped robot. We used the weighted outputs of mutually inhibited oscillators as torques to actuate joints. We verified the effectiveness of our proposal through simulation and compared the results to those of Taga’s CPG model, confirming better, more efficient generation of stable rhythmic walking at different speeds and robustness in response to disturbances.

Impulse pattern in bi-directionally coupled model neurons of different dynamics

The effects of bi-directional gap junction coupling of two model neurons with subthreshold oscillations have been examined when the individual neurons are operating at different dynamical states either in the tonic or bursting firing mode. Our simulations indicate that intermediate coupling strengths mostly lead to highly variable, often chaotic impulse patterns whereas transition to completely synchronized activity at high coupling strengths is generally going along with transitions to regular limit cycle activity. The synchronized activity pattern, however, can be completely different from the original pattern of the uncoupled neurons.

Noise-induced cooperative dynamics and its control in coupled neuron models

We investigate feedback control of the cooperative dynamics of two coupled neural oscillators that is induced merely by external noise. The interacting neurons are modeled as FitzHugh-Nagumo systems with parameter values at which no autonomous oscillations occur, and each unit is forced by its own source of random fluctuations. Application of delayed feedback to only one of two subsystems is shown to be able to change coherence and time scales of noise-induced oscillations either in the given subsystem, or globally. It is also able to induce or to suppress stochastic synchronization under certain conditions.

Bifurcation of synchronization phenomena observed in a coupled neuron model for circadian clock rhythms

Bifurcation of synchronization phenomena observed in a coupled neuron model for circadian clock rhythms

Influences of synaptic location on the synchronization of rhythmic bursting neurons

We study how the location of synaptic input influences the stablex firing states in coupled model neurons bursting rhythmically at the gamma frequencies (20-70 Hz). The model neuron consists of two compartments and generates one, two, three or four spikes in each burst depending on the intensity of input current and the maximum conductance of M-type potassium current. If the somata are connected by reciprocal excitatory synapses, we find strong correlations between the changes in the bursting mode and those in the stable phase-locked states of the coupled neurons. The stability of the in-phase phase-locked state (synchronously firing state) tends to change when the individual neurons change their bursting patterns. If, however, the synaptic connections are terminated on the dendritic compartments, no such correlated changes occur. In this case, the coupled bursting neurons do not show the in-phase phase-locked state in any bursting mode. These results indicate that synchronization behaviour of bursting neurons significantly depends on the synaptic location, unlike a coupled system of regular spiking neurons.

Influences of synaptic location on the synchronization of rhythmic bursting neurons

We study how the location of synaptic input influences the stablex firing states in coupled model neurons bursting rhythmically at the gamma frequencies (20–70 Hz). The model neuron consists of two compartments and generates one, two, three or four spikes in each burst depending on the intensity of input current and the maximum conductance of M-type potassium current. If the somata are connected by reciprocal excitatory synapses, we find strong correlations between the changes in the bursting mode and those in the stable phase-locked states of the coupled neurons. The stability of the in-phase phase-locked state (synchronously firing state) tends to change when the individual neurons change their bursting patterns. If, however, the synaptic connections are terminated on the dendritic compartments, no such correlated changes occur. In this case, the coupled bursting neurons do not show the in-phase phase-locked state in any bursting mode. These results indicate that synchronization behaviour of bursting neurons significantly depends on the synaptic location, unlike a coupled system of regular spiking neurons.

Anticipating the response of excitable systems driven by random forcing.

We study the regime of anticipated synchronization in unidirectionally coupled model neurons subject to a common external aperiodic forcing that makes their behavior unpredictable. We show numerically and by analog hardware electronic circuits that, under appropriate coupling conditions, the pulses fired by the slave neuron anticipate (i.e., predict) the pulses fired by the master neuron. This anticipated synchronization occurs even when the common external forcing is white noise.

ANN for transmission system static security assessment

This paper presents computationally efficient artificial neural network technique for assessing the security of the power system against line outages. Performance index (PI), which accounts for various line limit violations per contingency, is defined. The basic purpose of ANN is to assess the severity of line outages in terms of PI, based on training examples from off-line analysis. The selection of input signals for ANN is influenced by the operating state of the system and the contingency in question which determines the extent of line power limit violations. In an attempt to attain perfection in PI prediction, suitable architecture and topology for the network is investigated. To expedite learning process, saturating linear coupled neuron model (sl-CONE) is also tried out. The effectiveness of proposed technique is demonstrated on 5-bus (7-line) and IEEE 14-bus (21-line) test systems. Computation efficiency of the method makes it potential candidate for inclusion in on-line comprehensive security analysis package.

Noise shaping in populations of coupled model neurons

Biological information-processing systems, such as populations of sensory and motor neurons, may use correlations between the firings of individual elements to obtain lower noise levels and a systemwide performance improvement in the dynamic range or the signal-to-noise ratio. Here, we implement such correlations in networks of coupled integrate-and-fire neurons using inhibitory coupling and demonstrate that this can improve the system dynamic range and the signal-to-noise ratio in a population rate code. The improvement can surpass that expected for simple averaging of uncorrelated elements. A theory that predicts the resulting power spectrum is developed in terms of a stochastic point-process model in which the instantaneous population firing rate is modulated by the coupling between elements.

Traveling waves in a ring of three inhibitory coupled model neurons

The pyloric network of the stomatogastric ganglion of the lobster generates motor patterns with specific phase-lags between single neurons. This network inspired us to investigate a simplified model consisting of three mono-directionally coupled Morris–Lecar Oscillators. We have systematically analyzed the high-dimensional space of the synaptic parameters and identified parameter combinations which lead to biologically plausible phase-lags that exist even in a network with identical cells in the absence of an intrinsic burster. The dependence of the phase lags on the synaptic parameters was also explored.

Slow dynamics and regularization phenomena in ensembles of chaotic neurons

We have explored the role of calcium concentration dynamics in the generation of chaos and in the regularization of the bursting oscillations using a minimal neural circuit of two coupled model neurons. In regions of the control parameter space where the slowest component, namely the calcium concentration in the endoplasmic reticulum, weakly depends on the other variables, this model is analogous to three dimensional systems as found in [1] or [2]. These are minimal models that describe the fundamental characteristics of the chaotic spiking-bursting behavior observed in real neurons. We have investigated different regimes of cooperative behavior in large assemblies of such units using lattice of non-identical Hindmarsh-Rose neurons electrically coupled with parameters chosen randomly inside the chaotic region. We study the regularization mechanisms in large assemblies and the development of several spatio-temporal patterns as a function of the interconnectivity among nearest neighbors.