A stochastic model and a functional central limit theorem for information processing in large systems of neurons

The paper deals with information transmission in large systems of neurons. We model the membrane potential in a single neuron belonging to a cell tissue by a non time-homogeneous Cox-Ingersoll-Ross type diffusion; in terms of its time-varying expectation, this stochastic process can convey deterministic signals. We model the spike train emitted by this neuron as a Poisson point process compensated by the occupation time of the membrane potential process beyond the excitation threshold. In a large system of neurons 1≤i≤N processing independently the same deterministic signal, we prove a functional central limit theorem for the pooled spike train collected from the N neurons. This pooled spike train allows to recover the deterministic signal, up to some shape transformation which is explicit.     

Stochastic differential equation model for cerebellar granule cell excitability

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.     

神经元的确定性与随机性整数倍放电

在大鼠损伤背根节神经元的自发放电中发现了整数倍放电, 为了阐明这种放电所产生的原因, 首先研究神经元模型中确定性混沌所引起的整数倍放电与噪声所诱发的整数倍放电的峰峰间期（ＩＳＩ） 序列,通过分析得到前者的ＩＳＩ序列是非线性可预报的,具有确定的非线性特性,但由噪声所诱发的整数倍放电的ＩＳＩ序列是不可预报的, 这表明这两种机制所产生的整数倍放电具有不同的特点,存在着定性的差别,并且混沌运动所产生的整数倍放电是由混沌中各阶不稳定周期轨道决定的。从这种差别出发,分析了实验中整数倍放电的ＩＳＩ 序列,得到该ＩＳＩ 序列是可非线性预报的,这表明大鼠损伤背根节神经元自发放电中的整数倍放电更可能是由确定性机制所产生的     

The spatial properties of a model neuron increase its coding range

The coding properties of one-compartment and two-compartment model neurons are compared. The membrane depolarization in both models is described as a deterministic leaky integrator. Interspike intervals are identified with the periods between reset of the depolarization after firing and consecutive crossing of a fixed firing threshold. The two-point model has an input in the dendritic compartment and an output in the trigger-zone compartment. It is shown that the sensitivity threshold for the two-point model is shifted to the larger values of the input intensity with respect to the sensitivity threshold of its single-point counterpart. Further, its coding range is substantially larger than the coding range of the single-point model.     

Probabilistic Firing of Neurons Considered as a First Passage Problem

A formalized neuron receiving unitary excitatory impulses at random is considered. Each impulse provokes an effect of equal magnitude and of a duration not constant for each impulse, but which varies according to an exponential distribution. The effects sum until a threshold is reached when a response occurs. The distribution of intervals between successive responses is computed and compared with those obtained from a model in which the effects decay exponentially with time. Upon introducing inhibitory impulses also, the theory is applied to data on discharge characteristics of driven and spontaneously active thalamic neurons reported in the literature.     

Coefficient of variation of interspike intervals greater than 0.5. How and when?

Using Stein's model with and without reversal potentials, we investigated the mechanism of production of spike trains with a CV (ISI) (standard deviation/mean interspike interval) greater than 0.5, as observed in the visual cortex. When the attractor of the deterministic part of the dynamics is below the firing threshold, spike generation results primarily from random fluctuations. Using computer simulation for a range of membrane decay times and with other model parameters set to values appropriate for the visual cortex, we demonstrate that CV (ISI) is then usually greater than 0.5; if the attractor is above the threshold, spike generation is mainly due to deterministic forces, and CV (ISI) is then usually lower than 0.5. The critical value of the inhibitory postsynaptic potential (IPSP) rate at which CV (ISI) becomes greater than 0.5 is determined, resulting in specifications of how neurones might adjust their synaptic inputs to elicit irregular spike trains. Received: 25 June 1998/Accepted in revised form: 16 December 1998     

Variable initial depolarization in Stein's neuronal model with synaptic reversal potentials

The effect of a variable initial value is examined in Stein's stochastic neuronal model with synaptic reversal potentials under the conditions of a constant threshold and a constant input. The moments of the interspike interval distribution are presented as the functions of the initial depolarization which ranges from inhibitory reversal potential to the threshold potential. Normal, exponential and transformed Gamma distributions are tested for the initial value of depolarization. The coefficient of variation is shown to be greater than one when the initial depolarization is sufficiently above the resting level. An interpretation of this result in the terms of spatial facilitation is offered. The effect of a random initial value is found to be most pronounced for the neurons depolarized to a near threshold level.     

Amplification of trial-to-trial response variability by neurons in visual cortex

The visual cortex responds to repeated presentations of the same stimulus with high variability. Because the firing mechanism is remarkably noiseless, the source of this variability is thought to lie in the membrane potential fluctuations that result from summated synaptic input. Here this hypothesis is tested through measurements of membrane potential during visual stimulation. Surprisingly, trial-to-trial variability of membrane potential is found to be low. The ratio of variance to mean is much lower for membrane potential than for firing rate. The high variability of firing rate is explained by the threshold present in the function that converts inputs into firing rates. Given an input with small, constant noise, this function produces a firing rate with a large variance that grows with the mean. This model is validated on responses recorded both intracellularly and extracellularly. In neurons of visual cortex, thus, a simple deterministic mechanism amplifies the low variability of summated synaptic inputs into the large variability of firing rate. The computational advantages provided by this amplification are not known.     

Interspike Interval Fluctuations in Aplysia Pacemaker Neurons

In recent years, several mathematical models have been put forth to explain the time sequence of spike discharges in single neurons, in terms of synaptic inputs or intrinsic mechanisms. All of these models have been hypothetical, in that intracellular events were assumed, and not measured directly. The purpose of the present work was to study the statistics of the discharge from a preparation where intracellular recording was possible, and relate the observed discharge to measurable cell parameters. Regularly firing “pacemaker neurons” in the visceral ganglion of Aplysia californica were studied, using intracellular stimulating and recording techniques. Measurements were obtained of average curves of membrane potential, threshold for spike initiation, membrane resistance, and fluctuations of potential in the intervals between spontanously occurring spikes. The timing of discharges from these neurons was described quantitatively by interspike-interval histograms, mean and standard deviation of intervals, skewness, and serial correlation coefficients. A mathematical model (contained in a simulation program for the IBM 7094 computer) was constructed, based on discrete fluctuations of membrane potential following each spike and other directly observed intracellular events. It was found that the model could quantitatively account for observed spike trains, including variations in the discharge from one cell to another.     

A p-Adic model for the process of thinking disturbed by physiological and information noise

We develop a model of the process of thinking in the presence of noise (which is produced by the simultaneous action of a huge number of neurons in the brain as well as by external information and internal cognitive processes). Our model is based on Freud's idea on the splitting of cognitive processes into two (closely connected) domains: consciousness and subconsciousness. We represent the process of thinking as a random dynamical process in a space of ideas endowed with a non-Euclidean geometry (which differs extremely from the ordinary Euclidean geometry of spatial location of neurons in the brain). The so-called p-adic geometry on a space of ideas describes the ability of cognitive systems to form associations. We show that random dynamical thinking systems on a p -adic space of ideas still generate only deterministic ideas. We also study positive and negative effects of noise (in particular, creativeness and stress).     

The effect of a random initial value in neural first-passage-time models

The effect of a random initial value is examined in several stochastic integrate-and-fire neural models with a constant threshold and a constant input. The three models considered are approximations of Stein's model, namely: (1) a leaky integrator with deterministic trajectories, (2) a Wiener process with drift, and (3) an Ornstein-Uhlenbeck process. For model 1, different distributions for the initial value lead to commonly observed interspike interval distributions. For model 2, a discrete and a uniform distribution for the initial value are examined along with some parameter estimation procedures. For model 3, with a truncated normal distribution for the initial value, the coefficient of variation is shown to be greater than 1, and as the threshold becomes large the first-passage-time distribution approaches an exponential distribution. The relationships among the models and between them and previous models are also discussed, along with the robustness of the model assumptions and methods of their verification. The effects of a random initial value are found to be most pronounced at high firing rates.     

Diffusion approximation of the neuronal model with synaptic reversal potentials

The stochastic neuronal model with reversal potentials is approximated. For the model with constant postsynaptic potential amplitudes, a deterministic approximation is the only one which can be applied. The diffusion approximations are performed under the conditions of random postsynaptic potential amplitudes. New diffusion models of nerve membrane potential are devised in this way. These new models are more convenient for an analytical treatment than the original model with discontinuous trajectories.     

Method of evaluating the synaptic input to a single mesencephalic neuron]

The mathematical model of the spike activity of a neuron with synaptic input from many other neurons [1], describes adequately the firing of 5 from 7 neurons in the tegmentum of mesencephalic cat and changes of their activity evoked by glutamate iontophoresis. For these 5 neurons the estimates of the PSPs' average frequency of the threshold depolarization and of the constant decay of the EPSP were received. For different neurons the values of these parameters are 4--100 KHz, 100--800 average unitary EPSPs and 4--30 msec correspondingly. The stationary value of the average membrane potential (SVAMP) in all 5 neurons was removed significantly from the resting potential toward the threshold potential. SWAMP could be changed by the glutamate iontophoresis in such a degree to overlap the threshold potential.     

锥体神经元的连接模式对突触后局部电位的依赖

脑皮层的功能连接模式与突触可塑性密切相关,受突触空间分布和刺激模式等多种因素的影响。尽管越来越多的证据表明突触可塑性不仅受突触后动作电位而且还受突触后局部树突电位的影响,但是目前尚不清楚神经元的功能连接模式是否和怎样依赖于突触后局部电位的。为此,本文建立了一个无需硬边界设置的、突触后局部膜电位依赖的可塑性模型。该模型具有突触强度的自平衡能力并且能够再现多种突触可塑性实验结果。基于该模型对两个锥体神经元的功能连接模式进行仿真的结果表明,当突触后局部电位都处于亚阈值时两个神经元无功能连接,如果一个神经元的突触后膜电位高于阈值电位则产生向该神经元的单向连接,当两个神经元的突触后膜电位都超过阈值电位时则产生双向连接,说明突触后局部膜电位分布是神经元功能连接模式形成的关键。研究结果加深了神经网络连接模式形成机制的理解,对学习和记忆的研究具有重要意义。     

Optimal harvesting with exponential growth in an environment with random disasters and bonanzas

The optimal discounted present value of an exploited population under constant effort harvesting in an environment with random disasters and bonanzas is investigated. The deterministic component of growth is density independent (also called Malthusian or exponential). The disasters and bonanzas are random, occurring at the times of events of a Poisson process. The density independent properties of the model and the constant effort open loop policy lead to an exact solution for the expected present value. The optimal expected present value is compared with those in deterministic models with and without deterministic type jumps. Both deterministic and random jumps can have a significant influence on the optimal present value. However, the effort to achieve the optimal is not sensitive to variations in the total jump frequency or in the discount rate. The average random jump model is much easier to apply than the deterministic jump model. Bonanzas can have much more of an effect on the present value than disasters given similar jump rates.     

Contributions to the mathematical biophysics of the central nervous system with special reference to learning

A learning theory based on the lowering of thresholds of neurons under certain conditions is applied to two “random net” models. The first, a so-called “ganglion-brain” is characterized by completely random connections of all afferent tracts except certain ones which form the pathways for unconditioned responses. Certain expressions are derived which measure the learning potentiality of the ganglion— in particular, with respect to the number of responses which can be learned (conditioning potential) and the amount of interference between the learned responses (redundance potential). The second model concerns the progressive refinement of a response. The efficiency of learning in this case is reflected in the eventual specificity of the response which, in turn, depends on the modification of the distribution of thresholds associated with the neurons governing the responses. Expressions are derived relating the initial distribution of thresholds, the relative effectiveness of the various responses, and certain other parameters to the final distribution of thresholds. For a particular choice of the effectiveness distribution of responses the progressive sharpening of the threshold curve (i.e., progressive specificity of response) is demonstrated. Some implications of the model with respect to the evolution of nervous systems are discussed.     

A novel bursting mechanism of type a neurons in injured dorsal root ganglia

Using intracellular recording in vivo, the bursting behaviors were investigated in the neurons of chronically compressed dorsal root ganglia of the adult rat. In most cases, the first spike of a burst emerged from amplitude-increasing damped subthreshold membrane potential oscillation (SMPO) and the discharge terminated by an amplitude-decreasing damped SMPO. The rhythms of these bursting behaviors are all irregular. Since some researchers found that the stochastic dynamics can also produce similar bursting pattern, the deterministic dynamics of interevent interval (IEI) series obtained from raw membrane potential recording was detected by extraction of the hierarchy of unstable periodic orbits (UPOs) in the windowed IEI series. The results showed a number of statistically significant UPOs of order-one, order-two, and order-three. These orbits form a complex but predictable lattice of regions in which the dynamics of the bursting occurrence is deterministic. Based on a complete classification scheme, the investigated bursting can be depicted by the elliptic bursting dynamics. The significance of the finding that a neuron in the injured dorsal root ganglion has such dynamics is also discussed.     

Opiate slowing of feline respiratory rhythm and effects on putative medullary phase-regulating neurons

Opiates have effects on respiratory neurons that depress tidal volume and air exchange, reduce chest wall compliance, and slow rhythm. The most dose-sensitive opioid effect is slowing of the respiratory rhythm through mechanisms that have not been thoroughly investigated. An in vivo dose-response analysis was performed on medullary respiratory neurons of adult cats to investigate two untested hypotheses related to mechanisms of opioid-mediated rhythm slowing: 1) Opiates suppress intrinsic conductances that limit discharge duration in medullary inspiratory and expiratory neurons, and 2) opiates delay the onset and lengthen the duration of discharges postsynaptically in phase-regulating postinspiratory and late-inspiratory neurons. In anesthetized and unanesthetized decerebrate cats, a threshold dose (3 microg/kg) of the mu-opioid receptor agonist fentanyl slowed respiratory rhythm by prolonging discharges of inspiratory and expiratory bulbospinal neurons. Additional doses (2-4 microg/kg) of fentanyl also lengthened the interburst silent periods in each type of neuron and delayed the rate of membrane depolarization to firing threshold without altering synaptic drive potential amplitude, input resistance, peak action potential frequency, action potential shape, or afterhyperpolarization. Fentanyl also prolonged discharges of postinspiratory and late-inspiratory neurons in doses that slowed the rhythm of inspiratory and expiratory neurons without altering peak membrane depolarization and hyperpolarization, input resistance, or action potential properties. The temporal changes evoked in the tested neurons can explain the slowing of network respiratory rhythm, but the lack of significant, direct opioid-mediated membrane effects suggests that actions emanating from other types of upstream bulbar respiratory neurons account for rhythm slowing.