Bistability and resonance in the periodically stimulated Hodgkin-Huxley model with noise

On noise induced Poincaré-Andronov-Hopf bifurcation

It has been numerically seen that noise introduces stable well-defined oscillatory state in a system with unstable limit cycles resulting from subcritical Poincaré-Andronov-Hopf (or simply Hopf) bifurcation. This phenomenon is analogous to the well known stochastic resonance in the sense that it effectively converts noise into useful energy. Herein, we clearly explain how noise induced imperfection in the bifurcation is a generic reason for such a phenomenon to occur and provide explicit analytical calculations in order to explain the typical square-root dependence of the oscillations' amplitude on the noise level below a certain threshold value. Also, we argue that the noise can bring forth oscillations in average sense even in the absence of a limit cycle. Thus, we bring forward the inherent general mechanism of the noise induced Hopf bifurcation naturally realisable across disciplines.

Dynamics and Reliability of Bistable Neurons Driven with Time-Dependent Stimuli

The reliability of a spiking neuron depends on the frequency content of the driving input signal. Previous studies have shown that well above threshold, regularly firing neurons generate reliable responses when the input signal resonates with the firing frequency of the cell. Instead, well below threshold, reliable responses are obtained when the input frequency resonates with the subthreshold oscillations of the neuron. Previous theories, however, provide no clear prediction for the input frequency giving rise to maximally reliable spiking at threshold, which is probably the most relevant firing regime in mammalian cortex under physiological conditions. In particular, when the firing onset is governed by a subcritical Hopf bifurcation, the frequency of subthreshold oscillations often differs from the firing rate at threshold. The predictions of previous studies, hence, cannot be smoothly merged at threshold. Here we explore the behavior of reliability in bistable neurons near threshold using three types of driving stimuli: constant, periodic, and stochastic. We find that the two natural frequencies of the system, associated with the two coexisting attractors, provide a rich variety of possible locking modes with the external signal. Reliability is determined by the sensitivity to noise of each locking mode and by the transition probabilities between modes. Noise increases the amount of spike time jitter, and minimal jitter is obtained for input frequencies coinciding with the suprathreshold firing rate of the cell. In addition, noise may either enhance or inhibit transitions between the two attractors, depending on the input frequency. The dual role played by noise in bistable systems implies that reliability is determined by a delicate balance between spike time jitter and the rate of transitions between attractors.

Tunable oscillations in the Purkinje neuron

In this paper, we experimentally study the dynamics of slow oscillations in Purkinje neurons in vitro, and derive a strong association with a forced parametric oscillator model. We observed the precise rhythmicity of these oscillations in Purkinje neurons, as well as a dynamic tunability of this oscillation using a photoswitchable compound. We found that this slow oscillation can be induced in every Purkinje neuron measured, having periods ranging between 10 and 25 s. Starting from a Hodgkin-Huxley model, we demonstrate that this oscillation can be externally modulated, and that the neurons will return to their intrinsic firing frequency after the forced oscillation is concluded. These findings signify an additional timing functional role of tunable oscillations within the cerebellum, as well as a dynamic control of a time scale in the brain in the range of seconds.

Synchronization in a noise-driven developing neural network

We use computer simulations to investigate the structural and dynamical properties of a developing neural network whose activity is driven by noise. Structurally, the constructed neural networks in our simulations exhibit the small-world properties that have been observed in several neural networks. The dynamical change of neuronal membrane potential is described by the Hodgkin-Huxley model, and two types of learning rules, including spike-timing-dependent plasticity (STDP) and inverse STDP, are considered to restructure the synaptic strength between neurons. Clustered synchronized firing (SF) of the network is observed when the network connectivity (number of connections/maximal connections) is about 0.75, in which the firing rate of neurons is only half of the network frequency. At the connectivity of 0.86, all neurons fire synchronously at the network frequency. The network SF frequency increases logarithmically with the culturing time of a growing network and decreases exponentially with the delay time in signal transmission. These conclusions are consistent with experimental observations. The phase diagrams of SF in a developing network are investigated for both learning rules.