Diffusion approximation for a multi-input model neuron

A generalization of an earlier paper (Capocelli and Ricciardi, 1971), dealing with a diffusion approximation for a neuron subject to one excitatory and one inhibitory Poisson input, is provided by not imposing any restrictions on number and magnitude if synaptic inputs. An equation for the neuron's transition p.d.f. is derived, use of which is made to determine the moments of the membrane potential. It is finally shown that a diffusion approximation is possible and that the resulting diffusion process is characterized by constant infinitesimal variance and linear drift.     

Growth with regulation in random environment

The diffusion model for a population subject to Malthusian growth is generalized to include regulation effects. This is done by incorporating a logarithmic term in the regulation function in a way to obtain, in the absence of noise, an S-shaped growth law retaining the qualitative features of the logistic growth curve. The growth phenomenon is modeled as a diffusion process whose transition p.d.f. is obtained in closed form. Its steady state behavior turns out to be described by the lognormal distribution. The expected values and the mode of the transition p.d.f. are calculated, and it is proved that their time course is also represented by monotonically increasing functions asymptotically approaching saturation values. The first passage time problem is then considered. The Laplace transform of the first passage time p.d.f. is obtained for arbitrary thresholds and is used to calculate the expected value of the first passage time. The inverse Laplace transform is then determined for a threshold equal to the saturation value attained by the population size in the absence of random components. The probability of absorption for an arbitrary barrier is finally calculated as the limit of the absorption probability in a two-barrier problem.     

The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.     

An interaction model of a poisson and a renewal process related to neuron firing

This paper discusses a neuronal model based on a model of Coleman and Gastwirth (1969). It is assumed that the excitatory input forms a Poisson process while the inhibitory input forms a stationary renewal process. The proposed interaction scheme is as follows: an inhibitor deletes at most N consecutive excitatory inputs and a response only occurs after the cummalative storage of M excitatory inputs. The Laplace transform of the probability density function (p.d.f.) of the inter-response intervals is derived together with results of the numerical inversions.     

A computer program for the graphical and iterative fitting of probability density functions to biological data.

A tutorially-assisted, interactive program, written for a Digital Equipment Corporation LAB-11 minicomputer (PDP-11/20, is described which allows a user to fit (with or without automatic estimation of initial parameter values), by a method of nonlinear least squres, any one of seven different types of probability density functions (p.d.f.'s) to an empirical frequency distribution; the latter of which may be input to the program or formed by the program whenever it is furnished a series of times between events. The iteratively-obtained, "best fit" p.d.f. is displayed on a two color, point-plot display against the background of a point-plot histogram. By selecting any one of nine output modes, the user is allowed: (1) to view histograms successively on the point-plot display, (2) to generate selected p.d.f.'s (3) to "force" p.d.f.'s having known parameters through the histogram data, (4) to obtain Chi-square (x2) and Kolmogorov-Smionov estimates of the goodness of fit to the data, and (5) to apply a special test [Williams and Kloot, 1953] in order to determine whether the least squares estimates of two candidate models are statistically different. The resident driver program and the four overlayable program segments are written in standard FORTRAN IV; except for two plot routines, which are written in PDP-11 assembly language.     

Vaginal sonography of the cervix for the prediction of "time to delivery" in ART twins gestations.

The present study aimed to determine a reliable tool to estimate the interval time to delivery in assisted conception twin pregnancies. Mid-gestation cervical length was prospectively measured using transvaginal sonography (TVS) during routine antenatal care. Fifty-seven of 101 suitable women were longitudinally followed and two TVS measurements of their cervical length were obtained, first at approximately 24 weeks gestation and then at approximately 27 weeks gestation. The mean cervical length decreased from 37 +/- 12mm at first measurement to 34 +/- 11mm at the second one. A linear regression model was found between the time interval of the first (R = 0.656, p < 0.001) and the second (R = 0.435, p < 0.001) assessments and the week of delivery. The current data confirm that the length of the preserved segment of the cervix is an important indicator of its competence. A simple equation using the cervical length (mm) divided by 3 can predict mid gestation scan-to-delivery interval in twin gestation.     

A diffusion model for population growth in random environment

The growth of a population in a randomly varying environment is modeled by replacing the Malthusian growth rate with a delta-correlated normal process. The population size is then shown to be a random process, lognormally distributed, obeying a diffusion equation of the Fokker-Planck type. The first passage time p.d.f. through any arbitrarily assigned value and the probability of absorption are derived. The asymptotic behavior of the population size is investigated.     

Synaptic transmission in a model for stochastic neural activity.

A stochastic model equation for nerve membrane depolarization is derived which incorporates properties of synaptic transmission with a Rail-Eccles circuit for a trigger zone. If input processes are Poisson the depolarization is a Markov process for which equations for the moments of the interspike interval can be written down. An analytic result for the mean interval is obtained in a special case. The effect of the excitatory reversal potential is considerable if it is not too far from threshold and if the interspike interval is long. Computer simulations were performed when inhibitory and excitatory inputs are active. A substantial amount of inhibition leads to an exceedingly long tail in the density of the interspike time. With excitation only the interspike interval is often an approximately lognormal random variable. A coefficient of variation greater than one is often a consequence of relatively strong inhibition. Inferences can be made on the nature of the synaptic input from the statistics and density of the time between spikes. The inhibitory reversal potential usually has a relatively small effect except when the frequency of inhibition is large. An appendix contains the model equations in the case of an arbitrary distribution of postsynaptic potential amplitudes.     

The enhancement of stability of p53 in MTBP induced p53–MDM2 regulatory network

We have modeled an MTBP-MDM2–p53 regulatory network by integrating p53–MDM2 autoregulatory model (Proctor and Gray, 2008) with the effect of a cellular protein MTBP (MDM2 binding protein) which is allowed to bind with MDM2 (Brady et al., 2005). We study this model to investigate the activation of p53 and MDM2 steady state levels induced by MTBP protein under different stress conditions. Our simulation results in three approaches namely deterministic, Chemical Langevin equation and stochastic simulation of Master equation show a clear transition from damped limit cycle oscillation to fixed point oscillation during a certain time period with constant stress condition in the cell. This transition is the signature of transition of p53 and MDM2 levels from activated state to stabilized steady state levels. We present various phase diagrams to show the transition between unstable and stable states of p53 and MDM2 concentration levels and also their possible relations among critical value of the parameters at which the respective protein level reach stable steady states. In the stochastic approach, the dynamics of the proteins become noise induced process depending on the system size. We found that this noise enhances the stability of the p53 steady state level.     

Determination of the inter-spike times of neurons receiving randomly arriving post-synaptik potentials

Stein's model for a neuron receiving randomly arriving post-synaptic potentials is studied from an analytic viewpoint, using some recent results in the theory of first passage times for temporally homogeneous Markov processes. The case when the only input is excitatory can be treated exactly. It is shown that the moments of the firing time are guaranteed to be finite so that the differential-difference equation for the expectation (and higher moments) of the time for the membrane potential to first reach threshold from resting level can be written down. Analytic solutions are obtained in a number of cases with main emphasis on the case when the threshold is twice the epsp magnitude. An invariance principie is formulated wherein at a given mean input frequency and for a given decay parameter, the distribution of firing times depends only on the ratio of threshld to epsp magnitude. For the case where this ratio is two, the variation in the mean discharge rate is obtained as a function of mean input frequency. The results are compared with the experimental data for the Poisson monosynaptic excitation of cat motoneurons by Redmanet al. Agreement between theoretical and experimental values is excellent at input frequencies near 102 sec^-1, and theory underestimates the firing rate below that input frequency. Reasons for the discrepancy are discussed at length including the uncertainties in the neuronal parameters and the dependence of epsp magnitude on mean input frequency. The problem of including an inhibitory input process together with excitation is treated by an approximation procedure when the inhibition is considerably weaker than the excitation. At the input frequency investigated it is shown that when inhibition “half as weak” as the excitation occurs, the mean discharge frequency is approximately halved. In the final section a method of estimating neuronal parameters from the moments of the experimental inter-spike time distribution is outlined.     

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. Received: 5 March 2001 / Accepted in revised form: 7 August 2001     

Response characteristics of a neuron model to a periodic input

A mathematical neuron model defined by a difference equation was investigated when it was exposed to an environment of a periodic input stimulus. It is shown that pulse sequences constructed in advance by a particular method are actually realizable as the output of the system and the condition for the output sequence is also obtained with respect to the magnitude of the input. The results are interesting from a point of view of number theory.     

Entrainment in pacemakers characterized by a V-shaped PRC

The behaviour of a class of pacemakers characterized by a V-shaped PRC has been determined, for all possible frequencies and amplitudes of stimulation. The analytical study of the phase transition equation reveals that all rhythmic stimuli, but for a set of measure zero, give rise to entrainment. The ratio between firing and stimulation frequencies is a generalized Cantor function of the ratio between spontaneous and stimulation frequencies. A procedure to compute the detailed input/output pattern that underlies each entrainment ratio is given. Finally, the neurophysiological assumptions and implications of the results obtained are discussed.     

Autonomic energy conversion. I. The input relation: phenomenological and mechanistic considerations

The differences between completely and incompletely coupled linear energy converters are discussed using suitable electrochemical cells as examples. The output relation for the canonically simplest class of self-regulated incompletely coupled linear energy converters has been shown to be identical to the Hill force-velocity characteristic for muscle. The corresponding input relation (the “inverse” Hill equation) is now derived by two independent methods. The first method is a direct transformation of the output relation through the phenomenological equations of the converter; Onsager symmetry has no influence on the result. The second method makes use of a model system, a hydroelectric device with a regulator mechanism which depends only on the operational limits of the converter (an electro-osmosis cell operated in reverse) and on the load. The inverse Hill equation is shown to be the simplest solution of the regulator equation. An interesting and testable series of relations between input and output parameters arises from the two forms of the Hill equation. For optimal regulation the input should not be greatly different in the two limiting stationary states (level flow and static head). The output power will then be nearly maximal over a considerable range of load resistance, peak output being obtained at close to peak efficiency.     

A continuous cable method for determining the transient potential in passive dendritic trees of known geometry

Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.     

Firing rates of motoneurons with strong random synaptic excitation

The expected time to firing of a nerve impulse when there is Poisson excitation is calculated exactly in Stein's model. This is done at various input frequencies and various ratios of threshold to epsp magnitude, extending some previous calculations. The appropriate conditions for the validity of the model are discussed. Details of a particular calculation are given which involves the solution of a differential-difference equation. The results are presented as variation of expected time to firing as a function of input frequency for a given threshold to epsp ratio. The experimental results of Redman et al. for Poisson monosynaptic excitation of cat spinal motoneurons lead to the estimation of the epsp size which was not measured. The magnitude of the epsps predicted is in good agreement with that expected under the given conditions of stimulation. The predicted variation of epsp magnitude with input frequency is in accordance with that obtained in other experiments. When the finite rise time of epsps is taken into account the predicted epsp sizes are in better agreement with their expected amplitudes.     

Computation of spiking activity for a stochastic spatial neuron model: effects of spatial distribution of input on bimodality and CV of the ISI distribution

We obtain computational results for a new extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation and the synaptic input current is also a function of space and time, obeying a first order linear partial differential equation driven by a two-parameter random process. The model is first described explicitly with the inclusion of all biophysical parameters. Simplified equations are obtained with dimensionless space and time variables. A standard parameter set is described, based mainly on values appropriate for cortical pyramidal cells. When the noise is small and the mean voltage crosses threshold, a formula is derived for the expected time to spike. A simulation algorithm, involving one-dimensional random processes is given and used to obtain moments and distributions of the interspike interval (ISI). The parameters used are those for a near balanced state and there is great sensitivity of the firing rate around the balance point. This sensitivity may be related to genetically induced pathological brain properties (Rett's syndrome). The simulation procedure is employed to find the ISI distribution for some simple patterns of synaptic input with various relative strengths for excitation and inhibition. With excitation only, the ISI distribution is unimodal of exponential type and with a large coefficient of variation. As inhibition near the soma grows, two striking effects emerge. The ISI distribution shifts first to bimodal and then to unimodal with an approximately Gaussian shape with a concentration at large intervals. At the same time the coefficient of variation of the ISI drops dramatically to less than 1/5 of its value without inhibition.     

Derivation of a Quantitative Kinetic Model for a Visual Pigment from Observations of Early Receptor Potential

A “complete” and quantitative kinetic model for the states and transitions of the barnacle visual pigment in situ has been constructed from intracellular recordings of the early receptor potential responses to long light pulses. The model involves two stable and four thermolabile states and 10 photochemical, thermal, and metabolic transitions among them. The existence of each state and transition is demonstrated by qualitative examination of the response resulting from a carefully chosen experimental paradigm (combination of intensity, duration, and wavelength of adaptation and stimulation). Quantitative examination of the same responses determines all of the model transition rates, but only puts constraints on the state dipole moments. The latter are determined, and the former refined, by quantitative comparison of the predictions of the complete model with the responses to a set of paradigms chosen to involve as many states and transitions as possible. The fact that good fits can be obtained to these responses without further modification of the model supports its completeness.