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A simple stochastic mathematical model is developed and investigated for early human immunodeficiency virus type-1 (HIV-1) population dynamics. The model, which is a multi-dimensional diffusion process, includes activated uninfected CD4+T cells, latently and actively infected CD4+T cells and free virions occurring in plasma. Stochastic effects are assumed to arise in the process of infection of CD4+T cells and transitions may occur from uninfected to latently or actively infected cells by chance mechanisms. Using the best currently available parameter values, the intrinsic variability in response to a given initial infection is examined by solving the stochastic system numerically. We estimate the statistical distributions of the time of occurrence and the magnitude of the early peak in viral concentration. The maximum of the viral load has a value in the experimental range and its time of occurrence has a 95% confidence interval from 19.4 to 25.1 days. The stochastic nature of the growth of viral density is extremely pronounced in the first few days after initial infection. Threshold effects are noted at virion levels of about 3–5×10−5mm−3. In addition to modeling the intrinsic variability in HIV-1 growth, we have explored the effects of perturbations in the parameter values in order to assess the additional stochastic effects of between-patient variability. We found that changes in the initial number of virions or dose size, the rate at which latently infected CD4+T cells are converted to the actively infected form and the fraction of latent cells has only minor effects on the size, speed and variability of the response. In contrast, decreased speed and magnitude but greater variability in response were obtained when the death rate of uninfected CD4+T cells, the death rate of actively infected cells and the clearance rate of the virus were increased or when the appearance rate of uninfected CD4+T cells, the number of virions produced by infected cells, the infection rate of CD4+T cells and the initial number of uninfected activated CD4+T cells were decreased. We also determined the distribution of the time to reach a given virion density. From this distribution the probability of detection of the virus as a function of time can be estimated. The numerical results obtained are in the range of experimental values and are discussed in relation to recently proposed detection and testing procedures.

The depolarization of a passive nerve cylinder or dendritic tree in the equivalent cylinder representation is assumed to satisfy the cable equation. We consider in detail the effects of white noise current injection at a given location for the case of sealed end boundary conditions and for an initial resting state. The depolarization at a point is a Gaussian random process but is not Markovian. Expressions (infinite series) are obtained for the expectation, variance, spatial and temporal covariances of the depolarization. We examine the steady state expectation and variance and investigate how these are approached in time over the whole neuronal surface. We consider the relative contributions of various terms in the series for the expectation and variance of the depolarization at x=0 (soma, trigger zone, recording electrode) for various positions of the input process. It is found that different numbers of terms must be taken to obtain a reasonable approximation depending on whether the stimulus is at proximal, central or distal parts of the dendritic tree. We consider briefly the interspike time problem and see in an approximate way how spatial effects are important in determining the mean time between impulses.

The inhibitory influences of recurrent inhibition and afterhyperpolarization are studied theoretically insofar as they affect the density of the interspike interval and the frequency transfer characteristic. The methods employed involve exact results for excitation with decay and constant threshold, computer simulations for decaying thresholds representing afterhyperpolarization, and the diffusion approximation for excitation with inhibition and a constant threshold. Afterhyperpolarization tends to preserve the linearity of the frequency transfer characteristic and the lognormality of the interspike time. Recurrent inhibition which grows linearly with frequency of excitation can lead to frequency limiting. Some forms of nonlinear recurrent inhibition may lead to a filter type effect whereby the neuron responds significantly only over certain ranges of input intensity. A simple network model is analysed which exhibits recurrent inhibitory frequency growing linearly with frequency of excitation. An estimate of 10 to 50 is made for the number of Renshaw cells which connect with a spinal motoneuron. The frequency limiting of motoneurons is discussed and the stabilizing influence attributed to Renshaw cells is questioned. It is postulated that Renshaw recurrent inhibition is of functional significance at low levels of excitatory drive to motoneurons and that its effect is diminished by reciprocal inhibition at high excitatory input frequencies.

Exact cable theory is used to calculate voltage distributions along fully myelinated axons and those with various patterns of demyelination. The model employed uses an R-C circuit for the soma, an equivalent cable for the dendrites, a myelinated axon with n internodes and a cable representing telodendria. For the case of a voltage clamp at the soma, a system of 2n + 1 equations must be solved to obtain the potential distribution and this is done for arbitrary n. An explicit calculation is performed for one internode whereas computer-generated solutions are obtained for several internodes. The relative importance of the position of a single demyelinated internode is determined. An approximate expression is given for the critical internodal length necessary for action potential generation.

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

The usefulness of Green's function matrices for multi-dimensional diffusion equations is indicated. A general result is obtained and an application to two coupled cable equations is made.

Abstract.:  The growth of populations with continuous deterministic and random jump components is treated. Three special models in which random jumps occur at the time of events of a Poisson process and admit formal explicit solutions are considered: A) Logistic growth with random disasters having exponentially distributed amplitudes; B) Logistic growth with random disasters causing the removal of a uniformly distributed fraction of the population size; and C) Exponential decay with sudden increases (bonanzas) in the population and with each increase being an exponentially distributed fraction of the current population. Asymptotic and numerical methods are employed to determine the mean extinction time for the population, qualitatively and quantitatively. For Model A, this time becomes exponentially large as the carrying capacity becomes much larger than the mean disaster size. Implications for colonizing species for Model A are discussed. For Models B and C, the practical notion of a small, but positive, effective extinction level is chosen, and in these cases the expected extinction time rises rapidly with population size, yet at less than an e xponentially large order.

The coding of odor intensity by an olfactory receptor neuron model was studied under steady-state stimulation. Our model neuron is an elongated cylinder consisting of the following three components: a sensory dendritic region bearing odorant receptors, a passive region consisting of proximal dendrite and cell body, and an axon. First, analytical solutions are given for the three main physiological responses: (1) odorant-dependent conductance change at the sensory dendrite based on the Michaelis-Menten model, (2) generation and spreading of the receptor potential based on a new solution of the cable equation, and (3) firing frequency based on a Lapicque model. Second, the magnitudes of these responses are analyzed as a function of odorant concentration. Their dependence on chemical, electrical, and geometrical parameters is examined. The only evident gain in magnitude results from the activation-to-conductance conversion. An optimal encoder neuron is presented that suggests that increasing the length of the sensory dendrite beyond about 0.3 space constant does not increase the magnitude of the receptor potential. Third, the sensivities of the responses are examined as functions of (1) the concentration at half-maximum response, (2) the lower and upper concentrations actually discriminated, and (3) the width of the dynamic range. The overall gain in sensitivity results entirely from the conductance-to-voltage conversion. The maximum conductance at the sensory dendrite appears to be the main tuning constant of the neuron because it determines the shift toward low concentrations and the increase in dynamic range. The dynamic range of the model cannot exceed 5.7 log units, for a sensitivity increase at low odor concentration is compensated by a sensitivity decrease at high odor concentration.

A mathematical model for an olfactory receptor neuron is investigated. The physiological and anatomical background required for the construction of a mathematical model are explained. The model, which has been described previously, has three components, including the sensory dendrite on which are found the receptor proteins themselves, and others consisting of a passive cable leading to a trigger zone and axon. In the present paper, we pursue an analytical approach for determining the change in time of the receptor potential in the important case of a subthreshold square pulse of odorant stimulation delivered uniformly at the sensory dendrite. Then, the input current increases in time to its asymptotic value. This latter condition means that we can use a Green's function approach in order to obtain accurate representations for the solution for the entire length of the nerve cell. In the case of finite cables the solution is obtained as an infinite series which is shown to converge and can be easily used to find the depolarization at all space and time points of interest. A steady-state result is obtained directly by solving the relevant ordinary differential equation. For a semi-infinite cable an explicit expression is found for the voltage as a function of time and space variables involving a single integral. However, the exact expression follows from this for the steady-state result. The analytical results obtained are compared to numerical solutions and employed to investigate the effect of varying the position of the trigger zone and the electronic length of the neuron.

Using an integral equation of Darling and Siegert in conjunction with the backward Kolmogorov equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of a temporally homogeneous Markov process from a set in the phase space. The results, which are similar to those for diffusion processes, are used to find the expectation of the time between impulses of a Stein model neuron.

The 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.