Henry C. Tuckwell

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

n Abstractn n 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, Rn =∞, 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.n n

Quantitative aspects of L-type Ca2+ currents

Ca(2+) 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 Ca(v)1.1-Ca(v)1.4. L-type calcium currents play many fundamental roles in cellular dynamical processes including control of firing rate and 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 (I(CaL)) have been ascribed values as low as -50mV and as high as near 0mV. The inactivation of I(CaL) 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(CaL) 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(CaL) 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 complete 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 that are high, ≈ -18.3mV, or low, ≈ -36.4mV, which correspond closely with those for Ca(v)1.2 and Ca(v)1.3 channels in physiological solutions. There are very few complete experimental data on time constants of activation, those available suggesting values around 0.5-2ms. In modeling, a wide range of time constants has been employed. A major problem for quantitative studies due to lack of experimental data has been the use of kinetic parameters from one cell type for others. Inactivation time constants for VDI have been found experimentally with average 65ms. Examples of calculations of I(CaL) are made for linear and constant field methods and the effects of CDI are illustrated for single and double pulse protocols and the results compared with experiment. The review ends with a discussion and analysis of experimental subtype (Ca(v)1.1-Ca(v)1.4) properties and their roles in normal, including pacemaker, activity, and many pathological states.

Inhibition of rhythmic neural spiking by noise: the occurrence of a minimum in activity with increasing noise

The effects of noise on neuronal dynamical systems are of much current interest. Here, we investigate noise-induced changes in the rhythmic firing activity of single Hodgkin–Huxley neurons. With additive input current, there is, in the absence of noise, a critical mean value µu2009=u2009µc above which sustained periodic firing occurs. With initial conditions as resting values, for a range of values of the mean µ near the critical value, we have found that the firing rate is greatly reduced by noise, even of quite small amplitudes. Furthermore, the firing rate may undergo a pronounced minimum as the noise increases. This behavior has the opposite character to stochastic resonance and coherence resonance. We found that these phenomena occurred even when the initial conditions were chosen randomly or when the noise was switched on at a random time, indicating the robustness of the results. We also examined the effects of conductance-based noise on Hodgkin–Huxley neurons and obtained similar results, leading to the conclusion that the phenomena occur across a wide range of neuronal dynamical systems. Further, these phenomena will occur in diverse applications where a stable limit cycle coexists with a stable focus.

The probability of HIV infection in a new host and its reduction with microbicides

We use a simple mathematical model to estimate the probability and its time dependence that one or more HIV virions successfully infect target cells. For the transfer of a given number of virions to target cells we derive expressions for the probability P(inf), of infection. Thus, in the case of needlestick transfer we determine P(inf) and an approximate time window for post-exposure prophylaxis (PEP). For heterosexual transmission, where the transfer process is more complicated, a parameter gamma is employed which measures the strength of the infection process. For the smaller value of gamma, P(inf) is from 6 x 10(-5) to 0.93 or from 7.82 x 10(-6) to 0.29, where the lower figures are for the transfer of 100 virions and the upper figures are for the transfer of 4.4 million virions. We estimate the reductions in P(inf) which occur with a microbicide of a given efficacy. It is found that reductions may be approximately as stated when the number of virions transferred is less than about 10(5), but declines to zero for viral loads above that number. It is concluded that PEP should always be applied immediately after a needlestick incident. Further, manufacturers of microbicides should be encouraged to investigate and report their effectiveness at various transferred viral burdens.

Weak noise in neurons may powerfully inhibit the generation of repetitive spiking but not its propagation.

Many neurons have epochs in which they fire action potentials in an approximately periodic fashion. To see what effects noise of relatively small amplitude has on such repetitive activity we recently examined the response of the Hodgkin-Huxley (HH) space-clamped system to such noise as the mean and variance of the applied current vary, near the bifurcation to periodic firing. This article is concerned with a more realistic neuron model which includes spatial extent. Employing the Hodgkin-Huxley partial differential equation system, the deterministic component of the input current is restricted to a small segment whereas the stochastic component extends over a region which may or may not overlap the deterministic component. For mean values below, near and above the critical values for repetitive spiking, the effects of weak noise of increasing strength is ascertained by simulation. As in the point model, small amplitude noise near the critical value dampens the spiking activity and leads to a minimum as noise level increases. This was the case for both additive noise and conductance-based noise. Uniform noise along the whole neuron is only marginally more effective in silencing the cell than noise which occurs near the region of excitation. In fact it is found that if signal and noise overlap in spatial extent, then weak noise may inhibit spiking. If, however, signal and noise are applied on disjoint intervals, then the noise has no effect on the spiking activity, no matter how large its region of application, though the trajectories are naturally altered slightly by noise. Such effects could not be discerned in a point model and are important for real neuron behavior. Interference with the spike train does nevertheless occur when the noise amplitude is larger, even when noise and signal do not overlap, being due to the instigation of secondary noise-induced wave phenomena rather than switching the system from one attractor (firing regularly) to another (a stable point).

Analytical and simulation results for the stochastic spatial fitzhugh-nagumo model neuron

For the Fitzhugh-Nagumo system with space-time white noise, we use numerical methods to consider the generation of action potentials and the reliability of transmission in the presence of noise. The accuracy of simulated solutions is verified by comparison with known exact analytical results. Noise of small amplitude may prevent transmission directly, whereas larger-amplitude noise may also interfere by producing secondary nonlocal responses. The probability of transmission as a function of noise amplitude is found for both uniform noise and noise restricted to a patch. For certain parameter ranges, the recovery variable may be neglected to give a single-component nonlinear diffusion with space-time white noise. In this case, analytical results are obtained for small perturbations and noise, which agree well with simulation results. For the voltage variable, expressions are given for the mean, covariance, and variance and their steady-state forms. The spectral density of the voltage is also obtained. Numerical examples are given of the difference between the properties of nonlinear and linear cables, and the validity of the expressions obtained for the statistical properties is investigated as a function of noise amplitude. For given parameters, analytical results are in good agreement with simulation until a certain critical noise amplitude is reached, which can be estimated. The role of trigger zones in increasing the reliability of transmission is discussed.

The effects of various spatial distributions of weak noise on rhythmic spiking

We consider the response of the classical Hodgkin–Huxley (HH) spatial system in the weak to intermediate noise regime near the bifurcation to repetitive spiking. The deterministic component of the input (signal) is restricted to a small segment near the origin whereas noise, with parameter σ, occurs either only in the signal region or throughout the whole neuron. In both cases small noise inhibits the spiking and there is a minimum in the spike counts at σu2009≈u20090.15. At the same value of σ, the variance of the spike counts undergoes a pronounced maximum. For spatially restricted noise, the spike count continues to increase beyond the minimum until σu2009=u20090.5, but in the case of spatially extended noise the spike count begins to decline around σu2009=u20090.35 to give a local maximum. For both spatial distributions of noise, the variance of the spike count is found to also have a local minimum at about σu2009=u20090.4. Examples are given of the probability distributions of the spike counts and the spatial distributions of spikes with varying noise level. The differences in behaviours of the spike counts as noise increases beyond 0.3 are attributable to noise-induced spiking outside the signal region, which has a larger probability of occurrence when the noise is over an extended region. This aspect is investigated by ascertaining the probability of noise-induced spiking as a function of noise level and examination of the corresponding latency distributions. These findings prompt a definition of weak noise in the standard HH model as that for which the probability of secondary phenomena is negligible, which occurs when σ is less than about 0.3. Finally, if signal and weak (σu2009<u20090.3) noise are applied on disjoint intervals, then the noise has no effect on the instigation or propagation of spikes, no matter how large its region of application. These results are expected to apply to type 2 neurons in general, including the majority of cortical pyramidal cells.

Analysis of inverse stochastic resonance and the long-term firing of Hodgkin-Huxley neurons with Gaussian white noise

In order to explain the occurrence of a minimum in firing rate which occurs for certain mean input levels μ as noise level σ increases (inverse stochastic resonance, ISR) in Hodgkin–Huxley (HH) systems, we analyze the underlying transitions from a stable equilibrium point to limit cycle and vice-versa. For a value of μ at which ISR is pronounced, 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. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically. Long term trials of duration 500000 ms are carried out for values of σ from 0 to 2.0. The graph of mean spike count versus σ is divided into 4 regions R1,…,R4, where R3 contains the minimum associated with ISR. In R1 transitions to the basin of attraction of the rest point are not observed until a small critical value of σ=σc1 is reached, at the beginning of R2. The sudden decline in firing rate when σ is just greater than σc1 implies that there is only a small range of noise levels 0<σ<σc1 where repetitive spiking is safe from annihilation by noise. The firing rate remains small throughout R3. At a larger critical value σ=σc2 which signals the beginning of R4, the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle apparently becomes greater than zero and the spike rate thereafter increases with increasing σ. The quantitative scheme underlying the ISR curve is outlined in terms of the properties of exit time random variables. In the final subsection, several statistical properties of the main random variables associated with long term spiking activity are given, including distributions of exit times from the two relevant basins of attraction and the interspike interval.

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

Serotonergic neurons of the dorsal raphe nucleus, with their extensive innervation of limbic and higher brain regions and interactions with the endocrine system have important modulatory or regulatory effects on many cognitive, emotional and physiological processes. They have been strongly implicated in responses to stress and in the occurrence of major depressive disorder and other psychiatric disorders. In order to quantify some of these effects, detailed mathematical models of the activity of such cells are required which describe their complex neurochemistry and neurophysiology. 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 11 kinds of ion channels: a fast sodium current INa, a delayed rectifier potassium current IKDR, a transient potassium current IA, a slow non-inactivating potassium current IM, a low-threshold calcium current IT, two high threshold calcium currents IL and IN, small and large conductance potassium currents ISK and IBK, a hyperpolarization-activated cation current IH and a leak current ILeak. In Sections 3-8, each current type is considered in detail and parameters estimated from voltage clamp data where possible. Three kinds of model are considered for the BK current and two for the leak current. Intracellular calcium ion concentration Cai is an additional component and calcium dynamics along with buffering and pumping is discussed in Section 9. The remainder of the article contains descriptions of computed solutions which reveal both spontaneous and driven spiking with several parameter sets. Attention is focused on the properties usually associated with these neurons, particularly long duration of action potential, steep upslope on the leading edge of spikes, pacemaker-like spiking, long-lasting afterhyperpolarization and the ramp-like return to threshold after a spike. In some cases the membrane potential trajectories display doublets or have humps or notches as have been reported in some experimental studies. The computed time courses of IA and IT during the interspike interval support the generally held view of a competition between them in influencing the frequency of spiking. Spontaneous activity was facilitated by the presence of IH which has been found in these neurons by some investigators. For reasonable sets of parameters spike frequencies between about 0.6Hz and 1.2Hz are obtained, but frequencies as high as 6Hz could be obtained with special parameter choices. Topics investigated and compared with experiment include shoulders, notches, anodal break phenomena, the effects of noradrenergic input, frequency versus current curves, depolarization block, effects of cell size and the effects of IM. The inhibitory effects of activating 5-HT1A autoreceptors are also investigated. There is a considerable discussion of in vitro versus in vivo firing behavior, with focus on the roles of noradrenergic input, corticotropin-releasing factor and orexinergic inputs. Location of cells within the nucleus is probably a major factor, along with the state of the animal.

Stochastic Partial Differential Equations in Neurobiology: Linear and Nonlinear Models for Spiking Neurons

Stochastic differential equation (SDE) models of nerve cells for the most part neglect the spatial dimension. Including the latter leads to stochastic partial differential equations (SPDEs) which allow for the inclusion of important variations in the densities of ion channels. In the first part of this work, we briefly consider representations of neuronal anatomy in the context of linear SPDE models on line segments with one and two components. Such models are reviewed and analytical methods illustrated for finding solutions as series of Ornstein–Uhlenbeck processes. However, only nonlinear models exhibit natural spike thresholds and admit traveling wave solutions, so the rest of the article is concerned with spatial versions of the two most studied nonlinear models, the Hodgkin–Huxley system and the FitzHugh–Nagumo approximation. The ion currents underlying neuronal spiking are first discussed and a general nonlinear SPDE model is presented. Guided by recent results for noise-induced inhibition of spiking in the corresponding system of ordinary differential equations, in the spatial Hodgkin–Huxley model, excitation is applied over a small region and the spiking activity observed as a function of mean stimulus strength with a view to finding the critical values for repetitive firing. During spiking near those critical values, noise of increasing amplitudes is applied over the whole neuron and over restricted regions. Minima have been found in the spike counts which parallel results for the point model and which have been termed inverse stochastic resonance. A stochastic FitzHugh–Nagumo system is also described and results given for the probability of transmission along a neuron in the presence of noise.