Pulse-coupled System Research Articles

Noise-constrained switching times for heteroclinic computing.

Heteroclinic computing offers a novel paradigm for universal computation by collective system dynamics. In such a paradigm, input signals are encoded as complex periodic orbits approaching specific sequences of saddle states. Without inputs, the relevant states together with the heteroclinic connections between them form a network of states-the heteroclinic network. Systems of pulse-coupled oscillators or spiking neurons naturally exhibit such heteroclinic networks of saddles, thereby providing a substrate for general analog computations. Several challenges need to be resolved before it becomes possible to effectively realize heteroclinic computing in hardware. The time scales on which computations are performed crucially depend on the switching times between saddles, which in turn are jointly controlled by the system's intrinsic dynamics and the level of external and measurement noise. The nonlinear dynamics of pulse-coupled systems often strongly deviate from that of time-continuously coupled (e.g., phase-coupled) systems. The factors impacting switching times in pulse-coupled systems are still not well understood. Here we systematically investigate switching times in dependence of the levels of noise and intrinsic dissipation in the system. We specifically reveal how local responses to pulses coact with external noise. Our findings confirm that, like in time-continuous phase-coupled systems, piecewise-continuous pulse-coupled systems exhibit switching times that transiently increase exponentially with the number of switches up to some order of magnitude set by the noise level. Complementarily, we show that switching times may constitute a good predictor for the computation reliability, indicating how often an input signal must be reiterated. By characterizing switching times between two saddles in conjunction with the reliability of a computation, our results provide a first step beyond the coding of input signal identities toward a complementary coding for the intensity of those signals. The results offer insights on how future heteroclinic computing systems may operate under natural, and thus noisy, conditions.

Stochastic mean-field formulation of the dynamics of diluted neural networks

We consider pulse-coupled leaky integrate-and-fire neural networks with randomly distributed synaptic couplings. This random dilution induces fluctuations in the evolution of the macroscopic variables and deterministic chaos at the microscopic level [1]. Our main aim is to mimic the effect of the dilution as a noise source acting on the dynamics of a globally coupled nonchaotic system. We show that the evolution of a deterministic diluted neural network of any size can be well approximated by a much smaller fully coupled network, where each neuron is driven by a mean synaptic current plus additive noise. These terms represent the average and the fluctuations of the synaptic currents acting on the single neurons in the diluted system. The main microscopic and macroscopic dynamical features can be reproduced within this stochastic approximation. In order to illustrate the quality of this reconstruction, we compare the probability distribution function of the inter-spike intervals and the macroscopic attractor of the deterministic diluted system with those obtained by employing the mean-field stochastic model (see Figure ​Figure1A1A and ​and1B,1B, respectively). Furthermore, the microscopic stability of the diluted network can be also reproduced, as demonstrated from the almost coincidence of the measured Lyapunov exponents in the deterministic and stochastic cases for an ample range of system sizes (see Figure ​Figure1C.).1C.). Our results strongly suggest that the fluctuations in the synaptic currents are responsible for the emergence of chaos in this class of pulse-coupled systems [2]. Figure 1 A. Inter-spike interval distribution functions and B. attractor dynamics for a representative case of dilution with system size ND = 5000. C. Maximal Lyapunov exponent as a function of network size. In all panels, black circled symbols indicate the diluted ...

Renewal Approach to the Analysis of the Asynchronous State for Coupled Noisy Oscillators.

We develop a framework in which the activity of nonlinear pulse-coupled oscillators is posed within the renewal theory. In this approach, the evolution of the interevent density allows for a self-consistent calculation that determines the asynchronous state and its stability. This framework can readily be extended to the analysis of systems with more state variables and provides a population density treatment to evolve them in their thermodynamical limits. To demonstrate this we study a nonlinear pulse-coupled system, where couplings are dynamic and activity dependent. We investigate its stability and numerically study the nonequilibrium behavior of the system after the bifurcation. We show that this system undergoes a supercritical Hopf bifurcation to collective synchronization.

Pulse-coupled BZ oscillators with unequal coupling strengths.

Coupled chemical oscillators are usually studied with symmetric coupling, either between identical oscillators or between oscillators whose frequencies differ. Asymmetric connectivity is important in neuroscience, where synaptic strength inequality in neural networks commonly occurs. While the properties of the individual oscillators in some coupled chemical systems may be readily changed, enforcing inequality between the connection strengths in a reciprocal coupling is more challenging. We recently demonstrated a novel way of coupling chemical oscillators, which allows for manipulation of individual connection strengths. Here we study two identical, pulse-coupled Belousov-Zhabotinsky (BZ) oscillators with unequal connection strengths. When the pulse perturbations contain KBr (inhibitor), this system exhibits simple out-of-phase and complex oscillations, oscillatory-suppressed states as well as temporally periodic patterns (N : M) in which the two oscillators exhibit different numbers of peaks per cycle. The N : M patterns emerge due to the long-term effect of the inhibitory pulse-perturbations, a feature that has not been considered in earlier works. Time delay was previously shown to have a profound effect on the system's behaviour when pulse coupling was inhibitory and the coupling strengths were equal. When the coupling is asymmetric, however, delay produces no qualitative change in behaviour, though the 1 : 2 temporal pattern becomes more robust. Asymmetry in instantaneous excitatory coupling via AgNO3 injection produces a previously unseen temporal pattern (1 : N patterns starting with a double peak) with time delay and high [AgNO3]. Numerical simulations of the behaviour agree well with theoretical predictions in asymmetrical pulse-coupled systems.

Noise-induced Phenomena in Two Strongly Pulse-coupled Spiking Neurons

In this study, we address noise-induced phenomena in a system of two pulse-coupled resonate-and-fire neuron models as a piecewise smooth dynamical system. The model is a spiking neuron model that has second-order membrane dynamics with a firing threshold, and it exhibits subthreshold oscillation of membrane potential, resulting in the sensitivity to the timing of inputs. The pulse-coupled system shows the grazing bifurcation, a specific feature of a piecewise smooth dynamical system, as well as the saddle-node bifurcation. We numerically investigated mixture bifurcation structure of the system, leading to complex behaviors, such as burst synchronization and chaotic phenomena. Moreover, we found that a novel type of noise-induced order and chaos occur in the system when additive noise are given as a perturbation on the firing threshold. Finally, we will discuss a considerable mechanism of such noise-induced phenomena and the relationship to discontinuity-induced bifurcation in view of the piecewise smooth dynamical system theory.

Basic Dynamics of Simple Delay-Coupled Bifurcating Neurons

This letter studies a pulse-coupled system constructed by delayed cross-switching between two bifurcating neurons. The system can exhibit an interesting bifurcation: the delay-coupling can change chaotic behavior of single neurons into stable periodic behavior. Using the 1D phase map, it is clarified that the phenomenon is caused by the tangent bifurcation for the delay parameter. Presenting a simple test circuit, the phenomenon can be confirmed experimentally.

Controlled perturbation-induced switching in pulse-coupled oscillator networks

Pulse-coupled systems such as spiking neural networks exhibit nontrivial invariant sets in the form of attracting yet unstable saddle periodic orbits where units are synchronized into groups. Heteroclinic connections between such orbits may in principle support switching processes in these networks and enable novel kinds of neural computations. For small networks of coupled oscillators, we here investigate under which conditions and how system symmetry enforces or forbids certain switching transitions that may be induced by perturbations. For networks of five oscillators, we derive explicit transition rules that for two cluster symmetries deviate from those known from oscillators coupled continuously in time. A third symmetry yields heteroclinic networks that consist of sets of all unstable attractors with that symmetry and the connections between them. Our results indicate that pulse-coupled systems can reliably generate well-defined sets of complex spatiotemporal patterns that conform to specific transition rules. We briefly discuss possible implications for computation with spiking neural systems.

Rich dynamics of pulse-coupled spiking neurons with a triangular base signal

This paper presents a spiking neuron circuit with a triangular base signal. The circuit can output rich pulse-trains and the dynamics can be analyzed using a piecewise linear one-dimensional pulse-position map. Applying cross-switching to two neurons we construct a pulse-coupled system whose dynamics can be integrated into a composite map of the pulse position maps of two neurons. The composite map is piecewise linear and precise analysis is possible. We can clarify various interesting phenomena caused by the pulse-coupling. For example, periodic behavior of each neuron is changed into chaotic behavior and chaotic behavior of each neuron is changed into periodic behavior. These results provide basic information to construct flexible pulse-coupled neural networks.

Mutually pulse-coupled chaotic circuits by using dependent switched capacitors

This paper considers a mutually pulse-coupled system of chaotic circuits. Each circuit has a switch to short its capacitor, and the mutually pulse-coupling is realized by impulsive and simultaneous switching of the capacitors, depending on the states of both circuits. The coupled system exhibits synchronization of chaos and breakdown. The error expanding ratio between the units can be described by a simple theoretical formula. It enables us to classify the phenomena and to evaluate the dependence of the error expanding on the initial state. Fundamental bifurcation diagrams are given for the unit circuit and the coupled system. Some of the phenomena are confirmed by laboratory experiments.