Fast Threshold Modulation Research Articles

Possibilities of introducing different functional circuits on top of a structural neuron triplet: Where do the gains lie?

Instigated by the research on clusterization phenomena in complex neural networks, we study a triplet of bursting Rulkov map neurons connected via inhibitory synapses with delay. It is demonstrated how on a background of structural motif one can build different types of functional circuits. The approach is based on utilizing the properties of the chemical synapses, whose gating is modeled by the fast threshold modulation, in conjunction with the phase plane analysis, allowing the system state to be represented in terms of maps the neurons reside on. For both the dynamical configurations, monitoring the layout of active neurons, and the functional motifs, following the maps where the synchronized neurons lie, we establish a one-on-one correspondence between sequences in the time series and the triads, making up the subgraphs of the original graph. By introducing the appropriate sets of quantities, one obtains not only the distributions of triads as a function of synaptic parameters, but is also able to identify a distinct triad whose presence may be viewed as a signature of the burst synchronization process. In another setup, the regularization of burst cycles for an arbitrary neuron is explained by classifying all the bursts as long or short, with their fractions linked to the abundances of triads under variation of synaptic parameters.

Hybrid discrete-time neural networks

Hybrid dynamical systems combine evolution equations with state transitions. When the evolution equations are discrete-time (also called map-based), the result is a hybrid discrete-time system. A class of biological neural network models that has recently received some attention falls within this category: map-based neuron models connected by means of fast threshold modulation (FTM). FTM is a connection scheme that aims to mimic the switching dynamics of a neuron subject to synaptic inputs. The dynamic equations of the neuron adopt different forms according to the state (either firing or not firing) and type (excitatory or inhibitory) of their presynaptic neighbours. Therefore, the mathematical model of one such network is a combination of discrete-time evolution equations with transitions between states, constituting a hybrid discrete-time (map-based) neural network. In this paper, we review previous work within the context of these models, exemplifying useful techniques to analyse them. Typical map-based neuron models are low-dimensional and amenable to phase-plane analysis. In bursting models, fast-slow decomposition can be used to reduce dimensionality further, so that the dynamics of a pair of connected neurons can be easily understood. We also discuss a model that includes electrical synapses in addition to chemical synapses with FTM. Furthermore, we describe how master stability functions can predict the stability of synchronized states in these networks. The main results are extended to larger map-based neural networks.

Synchronization of bursting neurons with delayed chemical synapses

The synchronization of bursting Hindmarsh-Rose neurons coupled by a time-delayed fast threshold modulation synapse was studied. It is shown that there is a domain of the coupling parameter and nonzero time-lag values such that the bursting neurons are exactly synchronized. Furthermore, and contrary to the case of electrical synapses, such synchronous bursting is stochastically stable.

Bursting regimes in map-based neuron models coupled through fast threshold modulation

A system consisting of two map-based neurons coupled through reciprocal excitatory or inhibitory chemical synapses is discussed. After a brief explanation of the basic mechanism behind generation and synchronization of bursts, parameter space is explored to determine less obvious but biologically meaningful regimes and effects. Among them, we show how excitatory synapses without any delays may induce antiphase synchronization; that a synapse may change its character from excitatory to inhibitory and vice versa by changing its conductance, without any change in reversal potential; and that small variations in the synaptic threshold may result in drastic changes in the synchronization of spikes within bursts. Finally we show how the synchronization effects found in the two-neuron system carry over to larger networks.

On Synchronization and Traveling Waves in Chains of Relaxation Oscillators with an Application to Lamprey CPG

We study chains of relaxation-type neural oscillators with local excitatory coupling. Phase reductions suggest that such networks typically exhibit traveling waves, but relaxation oscillators often synchronize. We examine these behaviors using the phase response and fast threshold modulation (FTM) theories, which respectively describe network behavior for infinitesimally weak and moderate coupling. Surprisingly, the two different approximations yield quantitatively consistent predictions for chains with one-way coupling. Specifically, approaching the relaxation limit, such chains can exhibit waves with vanishing phase differences (i.e., synchrony) propagating in the coupling direction, or waves with persistent phase differences traveling against the coupling direction. These results provide novel support for the finding that caudo-rostral coupling dominates in the lamprey central pattern generator (CPG), and they suggest that recent models may underestimate the role of network effects in burst generation.

Synchronizing genetic relaxation oscillators by intercell signaling.

The ability to design and construct synthetic gene regulatory networks offers the prospect of studying issues related to cellular function in a simplified context; such networks also have many potential applications in biotechnology. A synthetic network exhibiting oscillatory behavior has recently been constructed [Elowitz, M. B. & Leibler, S. (2000) Nature (London) 403, 335-338]. It has also been shown that a natural bacterial quorum-sensing mechanism can be used in a synthetic system to communicate a signal between two populations of cells, such that receipt of the signal causes expression of a target gene [Weiss, R. & Knight, T. F. (2000) in DNA6: Sixth International Meeting on DNA-Based Computers, June 13-17, 2000, Leiden, The Netherlands]. We propose a synthetic gene network in Escherichia coli which combines these two features: the system acts as a relaxation oscillator and uses an intercell signaling mechanism to couple the oscillators and induce synchronous oscillations. We model the system and show that the proposed coupling scheme does lead to synchronous behavior across a population of cells. We provide an analytical treatment of the synchronization process, the dominant mechanism of which is "fast threshold modulation."

Phase Equations for Relaxation Oscillators

We use the Malkin theorem to derive phase equations for networks of weakly connected relaxation oscillators. We find an explicit formula for the connection functions when the oscillators have one-dimensional slow variables. The functions are discontinuous in the relaxation limit $\mu \rightarrow 0$, which provides a simple alternative illustration to the major conclusion of the fast threshold modulation (FTM) theory by Somers and Kopell [Biological Cybernetics, 68 (1993), pp. 393--407] that synchronization of relaxation oscillators has properties that are quite different from those of smooth (nonrelaxation) oscillators. We use Bonhoeffer--Van Der Pol relaxation oscillators to illustrate the theory numerically.

Waves and synchrony in networks of oscillators of relaxation and non-relaxation type

Arrays of relaxation oscillators behave differently in the presence of a non-uniformity of natural frequencies and/or coupling inputs than do arrays of phase oscillators coupled through phase differences. Phase oscillators compensate for non-uniformities by creating phase differences among the oscillators; relaxation oscillators coupled via “fast threshold modulation” (FTM) can respond by changing wave forms while leaving the fast jumps synchronous. For arrays of relaxation oscillators including chains, this allows synchrony to be a solution, even though the oscillators have different amounts of inputs from other oscillators; for phase-difference coupled oscillators in a chain, the generic solution is a travelling wave. Relaxation oscillators coupled through FTM also allow the encoding of patterns of oscillators into domains in which oscillators are in synchrony, with different domains in antiphase.

Rapid synchronization through fast threshold modulation.

Synchronization properties of locally coupled neural oscillators were investigated analytically and by computer simulation. When coupled in a manner that mimics excitatory chemical synapses, oscillators having more than one time scale (relaxation oscillators) are shown to approach synchrony using mechanisms very different from that of oscillators with a more sinusoidal waveform. The relaxation oscillators make critical use of fast modulations of their thresholds, leading to a rate of synchronization relatively independent of coupling strength within some basin of attraction; this rate is faster for oscillators that have conductance-based features than for neural caricatures such as the FitzHugh-Nagumo equations that lack such features. Computer simulations of one-dimensional arrays show that oscillators in the relaxation regime synchronize much more rapidly than oscillators with the same equations whose parameters have been modulated to yield a more sinusoidal waveform. We present a heuristic explanation of this effect based on properties of the coupling mechanisms that can affect the way the synchronization scales with array length. These results suggest that the emergent synchronization behavior of oscillating neural networks can be dramatically influenced by the intrinsic properties of the network components. Possible implications for perceptual feature binding and attention are discussed.