The space-clamped Hodgkin-Huxley system with random synaptic input: inhibition of spiking by weak noise and analysis with moment equations

We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model system obtained by an Euler method, it is found that with excitation only there is a critical value of the steady state excitatory conductance for repetitive spiking without noise and for values of the conductance near the critical value small noise has a powerfully inhibitory effect. For a given level of inhibition there is also a critical value of the steady state excitatory conductance for repetitive firing and it is demonstrated that noise either in the excitatory or inhibitory processes or both can powerfully inhibit spiking. Furthermore, near the critical value, inverse stochastic resonance was observed when noise was present only in the inhibitory input process. The system of 27 coupled deterministic differential equations for the approximate first and second order moments of the 6-dimensional model is derived. The moment differential equations are solved using Runge-Kutta methods and the solutions are compared with the results obtained by simulation for various sets of parameters including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex. The mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels. In the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells the assunptions underlying the moment equation approach are not satisfied

Computational modeling of spike generation in serotonergic neurons of the dorsal raphe nucleu

We consider here a single-compartment model of these neurons which is capable of describing many of the known features of spike generation, particularly the slow rhythmic pacemaking activity often observed in these cells in a variety of species. Included in the model are ten kinds of voltage dependent ion channels as well as calcium-dependent potassium current. Calcium dynamics includes buffering and pumping. In sections 3-9, each component is considered in detail and parameters estimated from voltage clamp data where possible. In the next two sections simplified versions of some components are employed to explore the effects of various parameters on spiking, using a systematic approach, ending up with the following eleven components: a fast sodium current $I_{Na}$, a delayed rectifier potassium current $I_{KDR}$, a transient potassium current $I_A$, a low-threshold calcium current $I_T$, two high threshold calcium currents $I_L$ and $I_N$, small and large conductance potassium currents $I_{SK}$ and $I_{BK}$, a hyperpolarization-activated cation current $I_H$, a leak current $I_{Leak}$ and intracellular calcium ion concentration $Ca_i$. Attention is focused on the properties usually associated with these neurons, particularly long duration of action potential, pacemaker-like spiking and the ramp-like return to threshold after a spike. In some cases the membrane potential trajectories display doublets or have kinks or notches as have been reported in some experimental studies. The computed time courses of $I_A$ and $I_T$ during the interspike interval support the generally held view of a competition between them in influencing the frequency of spiking. Spontaneous spiking could be obtained with small changes in a few parameters from their values with driven spiking.Comment: The abstract has been truncate

Random fluctuations at an equilibrium of a nonlinear reaction-diffusion equation

AbstractThe expectation of a dynamical variable satisfying a general nonlinear diffusion equation with small random fluctuations about an equilibrium point is found to order ϵ2. The dependence of the magnitude and direction of shift of the mean from the equilibrium on the properties of the nonlinear term is established