Coupled Van der Pol oscillators — A model of excitatory and inhibitory neural interactions

A system of mutually coupled Van der Pol equations is derived from an extended version of the Wilson and Cowan model for the dynamics of a number of excitatory and inhibitory neural subsets. In the lowest order of approximation, interactions between excitatory and inhibitory subsets appear as linear elastic coupling, while those within and between excitatory and excitatory subsets appear as nonlinear frictional coupling. The case of two coupled oscillators is investigated by the method of averaging and the stability conditions for two mode oscillations are obtained. Internal resonance is also discussed briefly in the case of identical oscillators.     

Analysis of stability of neural network with inhibitory neurons

Phase coding in a neural network composed of neural oscillators with inhibitory neurons was studied based on the theory of stochastic phase dynamics. We found that with increasing the coupling coefficients of inhibitory neural oscillators, the firing density in excitatory population transits to a critical state. In this case, when we increase the inhibitory coupling, the firing density will come into dynamic balance again and tend to a fixed value gradually. According to the phenomenon, in the paper we found parameter regions to exhibit those different population states, called dividing zones including flat fading zone, rapid fading zone and critical zone. Based on the dividing zones we can choose the number ratio between inhibitory neurons and excitatory neurons in the neural network, and estimate the coupling action of inhibitory population and excitatory population. Our research also shows that the balance value, enabling the firing density to reach the dynamic balance, does not depend on initial conditions. In addition, the critical value in critical state is only related to the number ratio between inhibitory neurons and excitatory neurons, but is independent of inhibitory coupling and excitatory coupling.     

Phase Resetting Curves Allow for Simple and Accurate Prediction of Robust N:1 Phase Locking for Strongly Coupled Neural Oscillators

Existence and stability criteria for harmonic locking modes were derived for two reciprocally pulse coupled oscillators based on their first and second order phase resetting curves. Our theoretical methods are general in the sense that no assumptions about the strength of coupling, type of synaptic coupling, and model are made. These methods were then tested using two reciprocally inhibitory Wang and Buzsáki model neurons. The existence of bands of 2:1, 3:1, 4:1, and 5:1 phase locking in the relative frequency parameter space was predicted correctly, as was the phase of the slow neuron's spike within the cycle of the fast neuron in which it occurred. For weak coupling the bands are very narrow, but strong coupling broadens the bands. The predictions of the pulse coupled method agreed with weak coupling methods in the weak coupling regime, but extended predictability into the strong coupling regime. We show that our prediction method generalizes to pairs of neural oscillators coupled through excitatory synapses, and to networks of multiple oscillatory neurons. The main limitation of the method is the central assumption that the effect of each input dies out before the next input is received.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Neural mechanisms potentially contributing to the intersegmental phase lag in lamprey

Factors contributing to the production of a phase lag along chains of oscillatory networks consisting of Hodgkin-Huxley type neurons are analyzed by means of simulations. Simplified network configurations are explored consisting of the basic building blocks of the spinal central pattern generator (CPG) generating swimming in the lamprey. It consists of reciprocally coupled crossed inhibitory C interneurons and ipsilateral excitatory E interneurons that activate C neurons and other E neurons. Oscillatory activity in the model network can, in the simplest case, be produced by a pair of reciprocally coupled C interneurons oscillating through an escape mechanism. Different levels of tonic excitation drive the network over a wide burst frequency range. In this type of network, powerful frequency-regulating factors are the effective inhibition produced by the active side, in combination with the tendency of the inactive side to escape from the inhibition. These two mechanisms can be affected by several factors, e.g. spike frequency adaptation (calcium-dependent K(+) channels), N-methyl-D-aspartate membrane properties as well as presence of low-voltage activated calcium channels. A rostrocaudal phase lag can be produced either by extending the contralateral inhibitory projections or the ipsilateral excitatory projections relatively more in the caudal than the rostral direction, since both an increased inhibition and a phasic excitation slow down the receiving network. The phase lag becomes decreased if the length of the intersegmental projections is increased or if the projections are extended symmetrically in both the rostral and the caudal directions. The simulations indicate that the conditions in the ends of an oscillator chain may significantly affect sign, magnitude and constancy of the phase lag. Also, with short and relatively weak intersegmental connections, the network remains robust against perturbations as well as intrinsic frequency differences along the chain. The phase lag (percentage of cycle duration) increases, however, with burst frequency also when the coupling strength is comparatively weak. The results are discussed and compared with previous "phase pulling" models as well as relaxation oscillators.     

Local and global synchronization transitions induced by time delays in small-world neuronal networks with chemical synapses

Effects of time delay on the local and global synchronization in small-world neuronal networks with chemical synapses are investigated in this paper. Numerical results show that, for both excitatory and inhibitory coupling types, the information transmission delay can always induce synchronization transitions of spiking neurons in small-world networks. In particular, regions of in-phase and out-of-phase synchronization of connected neurons emerge intermittently as the synaptic delay increases. For excitatory coupling, all transitions to spiking synchronization occur approximately at integer multiples of the firing period of individual neurons; while for inhibitory coupling, these transitions appear at the odd multiples of the half of the firing period of neurons. More importantly, the local synchronization transition is more profound than the global synchronization transition, depending on the type of coupling synapse. For excitatory synapses, the local in-phase synchronization observed for some values of the delay also occur at a global scale; while for inhibitory ones, this synchronization, observed at the local scale, disappears at a global scale. Furthermore, the small-world structure can also affect the phase synchronization of neuronal networks. It is demonstrated that increasing the rewiring probability can always improve the global synchronization of neuronal activity, but has little effect on the local synchronization of neighboring neurons.     

Analysis of clustered firing patterns in synaptically coupled networks of oscillators

Oscillators in networks may display a variety of activity patterns. This paper presents a geometric singular perturbation analysis of clustering, or alternate firing of synchronized subgroups, among synaptically coupled oscillators. We consider oscillators in two types of networks: mutually coupled, with all-to-all inhibitory connections, and globally inhibitory, with one excitatory and one inhibitory population of oscillators, each of arbitrary size. Our analysis yields existence and stability conditions for clustered states, along with formulas for the periods of such firing patterns. By using two different approaches, we derive complementary conditions, the first set stated in terms of time lengths determined by intrinsic and synaptic properties of the oscillators and their coupling and the second set stated in terms of model parameters and phase space structures directly linked to parameters. These results suggest how biological components may interact to produce the spindle sleep rhythm in thalamocortical networks. Received: 9 September 1999 / Revised version: 7 July 2000 / Published online: 24 November 2000     

Clustering in small networks of excitatory neurons with heterogeneous coupling strengths

Excitatory coupling with a slow rise time destabilizes synchrony between coupled neurons. Thus, the fully synchronous state is usually unstable in networks of excitatory neurons. Phase-clustered states, in which neurons are divided into multiple synchronized clusters, have also been found unstable in numerical studies of excitatory networks in the presence of noise. The question arises as to whether synchrony is possible in networks of neurons coupled through slow, excitatory synapses. In this paper, we show that robust, synchronous clustered states can occur in such networks. The effects of non-uniform distributions of coupling strengths are explored. Conditions for the existence and stability of clustered states are derived analytically. The analysis shows that a multi-cluster state can be stable in excitatory networks if the overall interactions between neurons in different clusters are stabilizing and strong enough to counter-act the destabilizing interactions between neurons within each cluster. When heterogeneity in the coupling strengths strengthens the stabilizing inter-cluster interactions and/or weakens the destabilizing in-cluster interactions, robust clustered states can occur in excitatory networks of all known model neurons. Numerical simulations were carried out to support the analytical results.     

Binding and segmentation of multiple objects through neural oscillators inhibited by contour information

 Temporal correlation of neuronal activity has been suggested as a criterion for multiple object recognition. In this work, a two-dimensional network of simplified Wilson-Cowan oscillators is used to manage the binding and segmentation problem of a visual scene according to the connectedness Gestalt criterion. Binding is achieved via original coupling terms that link excitatory units to both excitatory and inhibitory units of adjacent neurons. These local coupling terms are time independent, i.e., they do not require Hebbian learning during the simulations. Segmentation is realized by a two-layer processing of the visual image. The first layer extracts all object contours from the image by means of “retinal cells” with an “on-center” receptive field. Information on contour is used to selectively inhibit Wilson-Cowan oscillators in the second layer, thus realizing a strong separation among neurons in different objects. Accidental synchronism between oscillations in different objects is prevented with the use of a global inhibitor, i.e., a global neuron that computes the overall activity in the Wilson-Cowan network and sends back an inhibitory signal. Simulations performed in a 50×50 neural grid with 21 different visual scenes (containing up to eight objects + background) with random initial conditions demonstrate that the network can correctly segment objects in almost 100% of cases using a single set of parameters, i.e., without the need to adjust parameters from one visual scene to the next. The network is robust with reference to dynamical noise superimposed on oscillatory neurons. Moreover, the network can segment both black objects on white background and vice versa and is able to deal with the problem of “fragmentation.” The main limitation of the network is its sensitivity to static noise superimposed on the objects. Overcoming this problem requires implementation of more robust mechanisms for contour enhancement in the first layer in agreement with mechanisms actually realized in the visual cortex. Received: 25 October 2001 / Accepted: 26 February 2003 / Published online: 20 May 2003 Correspondence to: Mauro Ursino (e-mail: mursino@deis.unibo.it, Tel.: +39-051-2093008, Fax: +39-051-2093073)     

An Active Oscillator Model Describes the Statistics of Spontaneous Otoacoustic Emissions

Even in the absence of external stimulation, the cochleas of most humans emit very faint sounds below the threshold of hearing, sounds that are known as spontaneous otoacoustic emissions. They are a signature of the active amplification mechanism in the cochlea. Emissions occur at frequencies that are unique for an individual and change little over time. The statistics of a population of ears exhibit characteristic features such as a preferred relative frequency distance between emissions (interemission intervals). We propose a simplified cochlea model comprising an array of active nonlinear oscillators coupled both hydrodynamically and viscoelastically. The oscillators are subject to a weak spatial disorder that lends individuality to the simulated cochlea. Our model captures basic statistical features of the emissions: distributions of 1), emission frequencies; 2), number of emissions per ear; and 3), interemission intervals. In addition, the model reproduces systematic changes of the interemission intervals with frequency. We show that the mechanism for the preferred interemission interval in our model is the occurrence of synchronized clusters of oscillators.     

A cognitive and associative memory

By introducing a physiological constraint in the auto-correlation matrix memory, the system is found to acquire an ability in cognition i.e. the ability to identify and input pattern by its proximity to any one of the stored memories. The physiological constraint here is that the attribute of a given synapse (i.e. excitatory or inhibitory) is uniquely determined by the neuron it belongs. Thus the synaptic coupling is generally not symmetric. Analytical and numerical analyses revealed that the present model retrieves a memory if an input pattern is close to the pattern of the stored memories; if not, it gives a clear response by going into a special mode where almost all neurons are in the same state in each time step. This uniform mode may be stationary or periodic, depending on whether or not the number of the excitatory neurons exceeds the number of inhibitory neurons.     

Dynamics and bifurcations of two coupled neural oscillators with different connection types

In this paper we present an oscillatory neural network composed of two coupled neural oscillators of the Wilson-Cowan type. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. The network serves as a model for several possible network architectures. We study how the type and the strength of the connections between the oscillators affect the dynamics of the neural network. We investigate, separately from each other, four possible connection types (excitatory→excitatory, excitatory→inhibitory, inhibitory→excitatory, and inhibitory→inhibitory) and compute the corresponding bifurcation diagrams. In case of weak connections (small strength), the connection of populations of different types lead to periodicin-phase oscillations, while the connection of populations of the same type lead to periodicanti-phase oscillations. For intermediate connection strengths, the networks can enter quasiperiodic or chaotic regimes, and can also exhibit multistability. More generally, our analysis highlights the great diversity of the response of neural networks to a change of the connection strength, for different connection architectures. In the discussion, we address in particular the problem of information coding in the brain using quasiperiodic and chaotic oscillations. In modeling low levels of information processing, we propose that feature binding should be sought as a temporally coherent phase-locking of neural activity. This phase-locking is provided by one or more interacting convergent zones and does not require a central “top level” subcortical circuit (e.g. the septo-hippocampal system). We build a two layer model to show that although the application of a complex stimulus usually leads to different convergent zones with high frequency oscillations, it is nevertheless possible to synchronize these oscillations at a lower frequency level using envelope oscillations. This is interpreted as a feature binding of a complex stimulus.     

When inhibition not excitation synchronizes neural firing

Excitatory and inhibitory synaptic coupling can have counter-intuitive effects on the synchronization of neuronal firing. While it might appear that excitatory coupling would lead to synchronization, we show that frequently inhibition rather than excitation synchronizes firing. We study two identical neurons described by integrate-and-fire models, general phase-coupled models or the Hodgkin-Huxley model with mutual, non-instantaneous excitatory or inhibitory synapses between them. We find that if the rise time of the synapse is longer than the duration of an action potential, inhibition not excitation leads to synchronized firing.     

Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.     

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.     

Nonlinear equations of reaction-diffusion type for neural populations

Several simplified differential equations are derived from the Wilson and Cowan model describing the dynamics of excitatory and inhibitory neurons. It is shown, by expansions of the convolution integrals and the input-output functions, that the basic integrodifferential equations can be reduced to two coupled nonlinear partial differential equations of reaction-diffusion type. Further simplification leads to the coupled partial differential equations mathematically equivalent to the FitzHugh-Nagumo equations for the nerve impulse. Through a brief stability analysis in relation to the existing investigations on the bifurcation phenomena, an attempt is made to clarify the consequence due to the approximations introduced in this paper.     

Emergence of oscillations and spatio-temporal coherence states in a continuum-model of excitatory and inhibitory neurons

A neural field model of the reaction-diffusion type for the emergence of oscillatory phenomena in visual cortices is proposed. To investigate the joint spatio-temporal oscillatory dynamics in a continuous distribution of excitatory and inhibitory neurons, the coupling among oscillators is modelled as a diffusion process, combined with non-linear point interactions. The model exhibits cooperative activation properties in both time and space, by reacting to volleys of activations at multiple cortical sites with ordered spatio-temporal oscillatory states, similar to those found in the physiological experiments on slow-wave field potentials. The possible use of the resulting spatial distributions of coherent states, as a flexible medium to establish feature association, is discussed.     

Dynamics of spiking neurons connected by both inhibitory and electrical coupling

We study the dynamics of a pair of intrinsically oscillating leaky integrate-and-fire neurons (identical and noise-free) connected by combinations of electrical and inhibitory coupling. We use the theory of weakly coupled oscillators to examine how synchronization patterns are influenced by cellular properties (intrinsic frequency and the strength of spikes) and coupling parameters (speed of synapses and coupling strengths). We find that, when inhibitory synapses are fast and the electrotonic effect of the suprathreshold portion of the spike is large, increasing the strength of weak electrical coupling promotes synchrony. Conversely, when inhibitory synapses are slow and the electrotonic effect of the suprathreshold portion of the spike is small, increasing the strength of weak electrical coupling promotes antisynchrony (see Fig. 10). Furthermore, our results indicate that, given a fixed total coupling strength, either electrical coupling alone or inhibition alone is better at enhancing neural synchrony than a combination of electrical and inhibitory coupling. We also show that these results extend to moderate coupling strengths.