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Henry C Tuckwell

Publications and source records attributed to Henry C Tuckwell.

8 recordsLinked to original sources
PubMed2006

Cortical network modeling: analytical methods for firing rates and some properties of networks of LIF neurons.

The circuitry of cortical networks involves interacting populations of excitatory (E) and inhibitory (I) neurons whose relationships are now known to a large extent. Inputs to E- and I-cells may have their origins in remote or local cortical areas. We consider a rudimentary model involving E- and I-cells. One of our goals is to test an analytic approach to finding firing rates in neural networks without using a diffusion approximation and to this end we consider in detail networks of excitatory neurons with leaky integrate-and-fire (LIF) dynamics. A simple measure of synchronization, denoted by S(q), where q is between 0 and 100 is introduced. Fully connected E-networks have a large tendency to become dominated by synchronously firing groups of cells, except when inputs are relatively weak. We observed random or asynchronous firing in such networks with diverse sets of parameter values. When such firing patterns were found, the analytical approach was often able to accurately predict average neuronal firing rates. We also considered several properties of E-E networks, distinguishing several kinds of firing pattern. Included were those with silences before or after periods of intense activity or with periodic synchronization. We investigated the occurrence of synchronized firing with respect to changes in the internal excitatory postsynaptic potential (EPSP) magnitude in a network of 100 neurons with fixed values of the remaining parameters. When the internal EPSP size was less than a certain value, synchronization was absent. The amount of synchronization then increased slowly as the EPSP amplitude increased until at a particular EPSP size the amount of synchronization abruptly increased, with S(5) attaining the maximum value of 100%. We also found network frequency transfer characteristics for various network sizes and found a linear dependence of firing frequency over wide ranges of the external afferent frequency, with non-linear effects at lower input frequencies. The theory may also be applied to sparsely connected networks, whose firing behaviour was found to change abruptly as the probability of a connection passed through a critical value. The analytical method was also found to be useful for a feed-forward excitatory network and a network of excitatory and inhibitory neurons.

Action Potentials↗
PubMed2006

Some properties of a simple stochastic epidemic model of SIR type.

We investigate the properties of a simple discrete time stochastic epidemic model. The model is Markovian of the SIR type in which the total population is constant and individuals meet a random number of other individuals at each time step. Individuals remain infectious for R time units, after which they become removed or immune. Individual transition probabilities from susceptible to diseased states are given in terms of the binomial distribution. An expression is given for the probability that any individuals beyond those initially infected become diseased. In the model with a finite recovery time R, simulations reveal large variability in both the total number of infected individuals and in the total duration of the epidemic, even when the variability in number of contacts per day is small. In the case of no recovery, R=infinity, a formal diffusion approximation is obtained for the number infected. The mean for the diffusion process can be approximated by a logistic which is more accurate for larger contact rates or faster developing epidemics. For finite R we then proceed mainly by simulation and investigate in the mean the effects of varying the parameters p (the probability of transmission), R, and the number of contacts per day per individual. A scale invariant property is noted for the size of an outbreak in relation to the total population size. Most notable are the existence of maxima in the duration of an epidemic as a function of R and the extremely large differences in the sizes of outbreaks which can occur for small changes in R. These findings have practical applications in controlling the size and duration of epidemics and hence reducing their human and economic costs.

Algorithms↗
PubMed2004

On the behavior of solutions in viral dynamical models.

We consider simple mathematical models for the early population dynamics of the human immunodefficiency type 1 virus (HIV-1). Although these systems of differential equations may be solved by numerical methods, few general theoretical results are available due to nonlinearities. We analyze a model whose components are plasma densities of uninfected CD4+ T-cells and infected cells (assumed in this model to be proportional to virion density). In addition to analyzing the nature of the equilibrium points, we show that there are no periodic or limit-cycle solutions. Depending on the values of the parameters, solutions either tend without oscillation to an equilibrium point with zero virion density or to an equilibrium point in which there are a nonzero number of virions. In the latter case the approach to equilibrium may be through damped oscillations or without oscillation.

CD4-Positive T-Lymphocytes↗
PubMed2003

Optimal control of neuronal activity.

We investigate the optimal control of neuronal spiking activity for neurons receiving a class of random synaptic inputs, characterized by a positive parameter alpha. Optimal control signals and optimal variances are found exactly for the diffusion process approximating an integrate and fire model. When synaptic inputs are "sub-Poisson" (alpha<0.5), we find that the optimal synaptic input is a delta function (corresponding to bang-bang control) and the optimal signal is not unique. Poisson synaptic input is the critical case: The control signal is unique, but the control signal is still a delta function. For "supra-Poisson" (alpha>0.5) inputs, the optimal control is smooth and unique. The optimal variance obtained in the current paper sets the lowest possible bound in controlling the stochasticity of neuronal activity. We also discuss how to implement the optimal control signal for certain model neurons.

Action Potentials↗
PubMed2003

Determination of firing times for the stochastic Fitzhugh-Nagumo neuronal model.

We present for the first time an analytical approach for determining the time of firing of multicomponent nonlinear stochastic neuronal models. We apply the theory of first exit times for Markov processes to the Fitzhugh-Nagumo system with a constant mean gaussian white noise input, representing stochastic excitation and inhibition. Partial differential equations are obtained for the moments of the time to first spike. The observation that the recovery variable barely changes in the prespike trajectory leads to an accurate one-dimensional approximation. For the moments of the time to reach threshold, this leads to ordinary differential equations that may be easily solved. Several analytical approaches are explored that involve perturbation expansions for large and small values of the noise parameter. For ranges of the parameters appropriate for these asymptotic methods, the perturbation solutions are used to establish the validity of the one-dimensional approximation for both small and large values of the noise parameter. Additional verification is obtained with the excellent agreement between the mean and variance of the firing time found by numerical solution of the differential equations for the one-dimensional approximation and those obtained by simulation of the solutions of the model stochastic differential equations. Such agreement extends to intermediate values of the noise parameter. For the mean time to threshold, we find maxima at small noise values that constitute a form of stochastic resonance. We also investigate the dependence of the mean firing time on the initial values of the voltage and recovery variables when the input current has zero mean.

Models, Neurological↗
PubMed2002

A spatial stochastic neuronal model with Ornstein-Uhlenbeck input current.

We consider a spatial neuron model in which the membrane potential satisfies a linear cable equation with an input current which is a dynamical random process of the Ornstein-Uhlenbeck (OU) type. This form of current may represent an approximation to that resulting from the random opening and closing of ion channels on a neuron's surface or to randomly occurring synaptic input currents with exponential decay. We compare the results for the case of an OU input with those for a purely white-noise-driven cable model. The statistical properties, including mean, variance and covariance of the voltage response to an OU process input in the absence of a threshold are determined analytically. The mean and the variance are calculated as a function of time for various synaptic input locations and for values of the ratio of the time constant of decay of the input current to the time constant of decay of the membrane voltage in the physiological range for real neurons. The limiting case of a white-noise input current is obtained as the correlation time of the OU process approaches zero. The results obtained with an OU input current can be substantially different from those in the white-noise case. Using simulation of the terms in the series representation for the solution, we estimate the interspike interval distribution for various parameter values, and determine the effects of the introduction of correlation in the synaptic input stochastic process.

Action Potentials↗
PubMedDate not supplied

Firing properties of a stochastic PDE model of a rat sensory cortex layer 2/3 pyramidal cell.

We have developed a non-linear stochastic PDE (partial differential equation) model of a rat layer 2/3 somatosensory pyramidal neuron which approximates several of the dynamical properties of these cells. The model distinguishes telodendrites, a myelinated axon, initial segment, hillock, soma and a simplified dendritic tree. Distributions and properties of excitatory and inhibitory synapses were included, in accordance with recent anatomical and physiological findings. Using simulation methods, we aim to show that the spatial separation between regions of spatially distributed randomly activated excitatory and inhibitory synaptic inputs may be an important parameter which can influence neuronal firing properties. Due to the complexity of the problem, with respect to configurations of spatially and temporally activated excitatory and inhibitory synaptic inputs, we consider two simple configurations in which the spatial region of activated excitatory and inhibitory synaptic inputs overlap and when they are far from each. In the first, denoted configuration A, activated excitatory and inhibitory synapses were located close to the soma. In the second, denoted configuration B, active inhibitory synapses were close the soma, while active excitatory synapses were located on distal regions of the dendrite. For the first configuration, we find that increases in the mean rate of inhibition results in an increase in the width of the firing rate tuning curves, and that for particular mean input frequencies of excitation, increasing the mean input rate of inhibition does not always imply that the neuron fires at a slower rate. Furthermore, we observed for mean input frequencies of excitation between 15 and 60 (Hz), that increasing the mean rate of inhibition resulted in the linearization of the firing rate over this interval. For configuration B, no increase in width nor a linearization effect via inhibition was observed. These differences indicate that the distance between regions of active excitatory and inhibitory synapses may be an important factor to consider in determining how the interaction between excitation and inhibition contributes to neuronal firing.

Animals↗