Search

Serotonergic, noradrenergic and dopaminergic brainstem (including midbrain) neurons, often exhibit spontaneous and fairly regular spiking with frequencies of order a few Hz, though dopaminergic and noradrenergic neurons only exhibit such pacemaker-type activity in vitro or in vivo under special conditions. A large number of ion channel types contribute to such spiking so that detailed modeling of spike generation leads to the requirement of solving very large systems of differential equations. It is useful to have simplified mathematical models of spiking in such neurons so that, for example, features of inputs and output spike trains can be incorporated including stochastic effects for possible use in network models. In this article we investigate a simple two-component conductance-based model of the Hodgkin-Huxley type. Solutions are computed numerically and with suitably chosen parameters mimic features of pacemaker-type spiking in the above types of neurons. The effects of varying parameters is investigated in detail, it being found that there is extreme sensitivity to eight of them. Transitions from non-spiking to spiking are examined for two of these, the half-activation potential for an activation variable and the added (depolarizing) current and contrasted with the behavior of the classical Hodgkin-Huxley system. The plateaux levels between spikes can be adjusted, by changing a set of voltage parameters, to agree with experimental observations. Experiment has shown that in, in vivo, dopaminergic and noradrenergic neurons' pacemaker activity can be induced by the removal of excitatory inputs or the introduction of inhibitory ones. These properties are confirmed by mimicking opposite such changes in the model, which resulted in a change from pacemaker activity to bursting-type phenomena., Comment: arXiv admin note: substantial text overlap with arXiv:1508.05468

We investigate the dynamics of large-scale interacting neural populations, composed of conductance based, spiking model neurons with modifiable synaptic connection strengths, which are possibly also subjected to external noisy currents. The network dynamics is controlled by a set of neural population probability distributions ($\mathrm{PPD}$)which are constructed along the same lines as in the Klimontovich approach to the kinetic theory of plasmas. An exact non-closed, nonlinear, system of integro-partial differential equations is derived for the $\mathrm{PPD}$s. As is customary, a closing procedure leads to a mean field limit. The equations we have obtained are of the same type as those which have been recently derived using rigorous techniques of probability theory. The numerical solutions of these so called McKean-Vlasov-Fokker-Planck equations, which are only valid in the limit of infinite size networks, actually shows that the statistical measures as obtained from $\mathrm{PPD}$s are in good agreement with those obtained through direct integration of the stochastic dynamical system for large but finite size networks. Although numerical solutions have been obtained in the case of Fitzhugh-Nagumo model neurons, the theory can be readily applied to systems of Hodgkin-Huxley type model neurons of arbitrary dimension., Comment: 20 pages

Many cell populations, exemplified by certain tumors, grow approximately according to a Gompertzian growth model which has a slower approach to an upper limit than that of logistic growth. Certain populations of animals and other organisms have also recently been analyzed with the Gompertz model. This article addresses the question of how long it takes to reduce the population from one level to a lower one under a schedule of sudden decrements, each of which removes a given fraction of the cell mass or population. A deterministic periodic schedule is first examined and yields exact results for the eradication or extinction time which is defined as that required to reduce the number of cells to less than unity. The decrements in cell mass at each hit could correspond to an approximation to reduction of a tumor by external beam radiation therapy. The effects of variations in magnitude of successive decrements, the time interval between them, the initial population size and the intrinsic growth rate are calculated and results presented graphically. With a schedule governed by a Poisson process, the number of organisms or cells satisfies a stochastic differential equation whose solution sample paths have downward jumps as random times. The moments of the exit time then satisfy a system of recurrent differential-difference equations. A simple transformation results in a simpler system which has been studied both analytically and numerically in the context of interspike intervals of a model neuron. Results are presented for the mean eradication time for various frequencies and magnitudes of hits and for various eventual and initial population sizes. The standard deviation of the eradication time is also investigated.

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.

A simplified model for the growth of a population is studied in which random effects arise because reproducing individuals have a certain probability of surviving until the next breeding season and hence contributing to the next generation. The resulting Markov chain is that of a branching process with a known generating function. For parameter values leading to non-extinction, an approximating diffusion process is obtained for the population size. Results are obtained for the number of offspring $r_h$ and the initial population size $N_0$ required to guarantee a given probabilty of survival. For large probabilities of survival, increasing the initial population size from $N_0=1$ to $N_0=2$ gives a very large decrease in required fecundity but further increases in $N_0$ lead to much smaller decreases in $r_h$. For small probabilities (< 0.2) of survival the decreases in required fecundity when $N_0$ changes from 1 to 2 are very small. The calculations have relevance to the survival of populations derived from colonizing individuals which could be any of a variety of organisms.

Many central neurons, and in particular certain brainstem aminergic neurons exhibit spontaneous and fairly regular spiking with frequencies of order a few Hz. A large number of ion channel types contribute to such spiking so that accurate modeling of spike generation leads to the requirement of solving very large systems of differential equations, ordinary in the first instance. Since analysis of spiking behavior when many synaptic inputs are active adds further to the number of components, it is useful to have simplified mathematical models of spiking in such neurons so that, for example, stochastic features of inputs and output spike trains can be incorporated. In this article we investigate two simple two-component models which mimic features of spiking in serotonergic neurons of the dorsal raphe nucleus and noradrenergic neurons of the locus coeruleus. The first model is of the Fitzhugh-Nagumo type and the second is a reduced Hodgkin-Huxley model. For each model solutions are computed with two representative sets of parameters. Frequency versus input currents are found and reveal Hodgkin type 2 behavior. For the first model a bifurcation and phase plane analysis supports these findings. The spike trajectories in the second model are very similar to those in DRN SE pacemaker activity but there are more parameters than in the Fitzhugh-Nagumo type model. The article concludes with a brief review of previous modeling of these types of neurons and its relevance to studies of serotonergic involvement in spatial working memory and obsessive-compulsive disorder., Comment: 21 pages, 11 figures

We consider linear and nonlinear hyperbolic SPDEs with mixed derivatives with additive space-time Gaussian white noise of the form $Y_{xt}=F(Y) + \sigma W_{xt}.$ Such equations, which transform to linear and nonlinear wave equations, including Klein-Gordon, Liouville's and the sine-Gordon equation, are related to what Zimmerman (1972) called a diffusion equation. An explicit numerical scheme is employed in both deterministic and stochastic examples. The scheme is checked for accuracy against known exact analytical solutions for deterministic equations. In the stochastic case with $F=0$, solutions yield sample paths for the Brownian sheet whose statistics match well exact values. Generally the boundary conditions are chosen to be initial values $Y(x,0)$ and boundary values $Y(0,t)$ on the quarter-plane or subsets thereof, which have been shown to lead to existence and uniqueness of solutions. For the linear case solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms with no signal. Equations considered with nonlinear $F$ included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with $\sigma \le 0.25$ but they tended to be shattered at larger values of $\sigma$. Previous work on stochastic sine-Gordon equations is briefly reviewed.

Serotonergic neurons of the dorsal raphe nuclei, with their extensive innervation of nearly the whole brain have important modulatory effects on many cognitive and physiological processes. They play important roles in clinical depression and other psychiatric disorders. In order to quantify the effects of serotonergic transmission on target cells it is desirable to construct computational models and to this end these it is necessary to have details of the biophysical and spike properties of the serotonergic neurons. Here several basic properties are reviewed with data from several studies since the 1960s to the present. The quantities included are input resistance, resting membrane potential, membrane time constant, firing rate, spike duration, spike and afterhyperpolarization (AHP) amplitude, spike threshold, cell capacitance, soma and somadendritic areas. The action potentials of these cells are normally triggered by a combination of sodium and calcium currents which may result in autonomous pacemaker activity. We here analyse the mechanisms of high-threshold calcium spikes which have been demonstrated in these cells the presence of TTX. The parameters for calcium dynamics required to give calcium spikes are quite different from those for regular spiking which suggests the involvement of restricted parts of the soma-dendritic surface as has been found, for example, in hippocampal neurons.

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 truncated

In previous articles we have investigated the firing properties of the standard Hodgkin-Huxley (HH) systems of ordinary and partial differential equations in response to input currents composed of a drift (mean) and additive Gaussian white noise. For certain values of the mean current, as the noise amplitude increased from zero, the firing rate exhibited a minimum and this phenomenon was called inverse stochastic resonance (ISR). Here we analyse the underlying transitions from a stable equilibrium point to the limit cycle and vice-versa. Focusing on the case of a mean input current density $\mu=6.8$ at which repetitive firing occurs and ISR had been found to be pronounced, some of the properties of the corresponding stable equilibrium point are found. A linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur. A one dimensional diffusion is also constructed for small noise based on the correlations between the pairs of HH variables and the small magnitudes of the fluctuations in two of them. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically and also the nature of distribution of spikes at very small noise corresponding to trajectories which do not ever enter the basin of attraction of the equilibrium point. Long term trials of duration 500000 ms are carried out for values of the noise parameter $\sigma$ from 0 to 2.0, with results appearing in Section 3. The graph of mean spike count versus $\sigma$ is divided into 4 regions $R_1,...,R_4,$ where $R_3$ contains the minimum associated with ISR., Comment: 27 pages, 16 figures

We consider the standard three-component differential equation model for the growth of an HIV virion population in an infected host in the absence of drug therapy. The dynamical properties of the model are determined by the set of values of six parameters which vary across host populations. There may be one or two critical points whose natures play a key role in determining the outcome of infection and in particular whether the HIV population will persist or become extinct. There are two cases which may arise. In the first case, there is only one critical point P_1 at biological values and this is an asymptotically stable node. The system ends up with zero virions and so the host becomes HIV-free. In the second case, there are two critical points P_1 and P_2 at biological values. Here P_1 is an unstable saddle point and P_2 is an asymptotically stable spiral point with a non-zero virion level. In this case the HIV population persists unless parameters change. We let the parameter values take random values from distributions based on empirical data, but suitably truncated, and determine the probabilities of occurrence of the various combinations of critical points. From these simulations the probability that an HIV infection will persist, across a population, is estimated. It is found that with conservatively estimated distributions of parameters, within the framework of the standard 3-component model, the chances that a within host HIV population will become extinct is between 0.6% and 6.9%. With less conservative parameter estimates, the probability is estimated to be as high as 24%. The many factors related to the transmission and possible spontaneous elimination of the virus are discussed., Comment: 19 pages, 4 figures

Voltage clamp data were analyzed in order to characterize the properties of the fast potassium transient current IA for a serotonergic neuron of the rat dorsal raphe nucleus (DRN). We obtain maximal conductance, time constants of activation and inactivation, and the steady state activation and inactivation functions, as Boltzmann curves, defined by half-activation potentials and slope factors. We employ a novel method to accurately obtain the activation function and compare the results with those obtained by other methods. The form of IA is estimated as g(V-V_{rev}) m^4h with g=20.5 nS. For activation, the half-activation potential is V_a=-52.5 mV with slope factor k_a=16.5 mV, whereas for inactivation the corresponding quantities are -91.5 mV and -9.3 mV. We discuss the results in terms of the corresponding properties of \IA in other cell types and their possible relevance to pacemaking activity in 5-HT cells of the DRN., Comment: 16 pages, 4 figures

Calcium currents in neurons and muscle cells have been classified as being one of 5 types of which four, L, N, P/Q and R were said to be high threshold and one, T, was designated low threshold. This review focuses on quantitative aspects of L-type currents. L-type channels are now distinguished according to their structure as one of four main subtypes 1.1-1.4. L-type calcium currents play many fundamental roles in cellular dynamical processes including pacemaking in neurons and cardiac cells, the activation of transcription factors involved in synaptic plasticity and in immune cells. The half-activation potentials of L-type currents have been ascribed values as low as -50 mV and as high as near 0 mV. The inactivation of I_L has been found to be both voltage (VDI) and calcium-dependent (CDI) and the latter component may involve calcium-induced calcium release. CDI is often an important aspect of dynamical models of cell electrophysiology. We describe the basic components in modeling I_L including activation and both voltage and calcium dependent inactivation and the two main approaches to determining the current. We review, by means of tables of values from over 65 representative studies, the various details of the dynamical properties associated with I_L that have been found experimentally or employed in the last 25 years in deterministic modeling in various nervous system and cardiac cells. Distributions and statistics of several parameters related to activation and inactivation are obtained. There are few reliable experimental data on L-type calcium current kinetics for cells at physiological calcium ion concentrations. Neurons are divided approximately into two groups with experimental half-activation potentials., Comment: 92 pages 12 figures

Neurons in the central nervous system are affected by complex and noisy signals due to fluctuations in their cellular environment and in the inputs they receive from many other cells 1,2. Such noise usually increases the probability that a neuron will send out a signal to its target cells 2-5. In stochastic resonance, which occurs in many physical and biological systems, an optimal response is found at a particular noise amplitude 6-9. We have found that in a classical neuronal model the opposite can occur - that noise can subdue or turn off repetitive neuronal activity in both single cells and networks of cells. Recent experiments on regularly firing neurons with noisy inputs confirm these predictions 10,11. Surprisingly, we find that in some cases there is a noise level at which the response is a minimum, a phenomenon which is called inverse stochastic resonance. Suppression of rhythmic behavior by noise and inverse stochastic resonance are predicted to occur not only in neuronal systems but more generally in diverse nonlinear dynamical systems where a stable limit cycle is attainable from a stable rest state., Comment: 12 pages, 4 figures

We examine the effects of stochastic input currents on the firing behavior of two excitable neurons coupled with fast excitatory synapses. In such cells (models), typified by the quadratic integrate and fire model, mutual synaptic coupling can cause sustained firing or oscillatory behavior which is necessarily antiphase. Additive Gaussian white noise can transiently terminate the oscillations, hence destroying the stable limit cycle. Further application of the noise may return the system to spiking activity. In a particular noise range, the transition times between the oscillating and the resting state are strongly asymmetric. We numerically investigate an approximate basin of attraction, A, of the periodic orbit and use Markov process theory to explain the firing behavior in terms of the probability of escape of trajectories from A

The use of cortical field potentials rather than the details of spike trains as the basis for cognitive information processing is proposed. This results in a space of cognitive elements with natural metrics. Sets of spike trains may also be considered to be points in a multidimensional metric space. The closeness of sets of spike trains in such a space implies the closeness of points in the resulting function space of potential distributions., Comment: 4 pages