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.     

Ionic and synaptic mechanisms underlying a brainstem oscillator: An in vitro study of the pacemaker nucleus of Apteronotus

1. In an in vitro preparation of the medullary pacemaker nucleus of Apteronotus, the consequences of a variety of ionic and pharmacological manipulations upon both ongoing activity and synaptic modulation of the nucleus were assessed. 2. Spontaneous rhythmicity in the pacemaker nucleus was found to be Na+-, K+-, and Ca(2+)-dependent. The extreme sensitivity to 4-aminopyridine (4-AP) relative to other treatments suggested that the K+ A-current is a critical element in the oscillations. 3. Elevated K+ or 4-AP were titrated to concentrations that suppressed spontaneous oscillations, but allowed modulatory, 'chirp' epsps to persist. The transition to elevated K+ revealed oscillatory properties in some neurons in the form of epsp-induced ringing 4. Threshold concentrations of 4-AP sufficient to halt oscillations, caused epsps to become larger and complex, increased input resistance, and enhanced the effects of current injection on epsp amplitude. A greater degree of voltage-sensitivity was also seen in later components of the complex epsp. 5. Several treatments presumed to increase Ca2+ caused desynchronization of firing and revealed diverging intrinsic frequencies among cells.     

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.     

Firing rates of neurons with random excitation and inhibition

The expectation of the interspike interval for a Stein model neuron receiving Poisson excitation and inhibition is determined by solving a differential difference equation with both forward and backward differences. The method of solution relies on an asymptotic expansion at large initial hyperpolarizations. The asymptotic solution is continued to near threshold depolarization whereupon the boundary condition is employed along with recursion relations to obtain the complete solution. The dependency of the mean firing rate on excitation at fixed inhibition and on inhibition at fixed excitation is investigated as well as the threshold dependence at fixed input rates. The results are discussed in relation to those for intracellular current injection and synaptic input to real neurons.     

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.     

Influence of temporal correlation of synaptic input on the rate and variability of firing in neurons

The spike trains that transmit information between neurons are stochastic. We used the theory of random point processes and simulation methods to investigate the influence of temporal correlation of synaptic input current on firing statistics. The theory accounts for two sources for temporal correlation: synchrony between spikes in presynaptic input trains and the unitary synaptic current time course. Simulations show that slow temporal correlation of synaptic input leads to high variability in firing. In a leaky integrate-and-fire neuron model with spike afterhyperpolarization the theory accurately predicts the firing rate when the spike threshold is higher than two standard deviations of the membrane potential fluctuations. For lower thresholds the spike afterhyperpolarization reduces the firing rate below the theory's predicted level when the synaptic correlation decays rapidly. If the synaptic correlation decays slower than the spike afterhyperpolarization, spike bursts can occur during single broad peaks of input fluctuations, increasing the firing rate over the prediction. Spike bursts lead to a coefficient of variation for the interspike intervals that can exceed one, suggesting an explanation of high coefficient of variation for interspike intervals observed in vivo.     

Computational Consequences of Temporally Asymmetric Learning Rules: II. Sensory Image Cancellation

The electrosensory lateral line lobe (ELL) of mormyrid electric fish is a cerebellum-like structure that receives primary afferent input from electroreceptors in the skin. Purkinje-like cells in ELL store and retrieve a temporally precise negative image of prior sensory input. The stored image is derived from the association of centrally originating predictive signals with peripherally originating sensory input. The predictive signals are probably conveyed by parallel fibers. Recent in vitro experiments have demonstrated that pairing parallel fiber-evoked excitatory postsynaptic potentials (epsps) with postsynaptic spikes in Purkinje-like cells depresses the strength of these synapses. The depression has a tight dependence on the temporal order of pre- and postsynaptic events. The postsynaptic spike must follow the onset of the epsp within a window of about 60 msec for the depression to occur and pairings at other delays yield a nonassociative enhancement of the epsp. Mathematical analyses and computer simulations are used here to test the hypothesis that synaptic plasticity of the type established in vitro could be responsible for the storage of temporal patterns that is observed in vivo. This hypothesis is confirmed. The temporally asymmetric learning rule established in vitro results in the storage of activity patterns as observed in vivo and does so with significantly greater fidelity than other types of learning rules. The results demonstrate the importance of precise timing in pre- and postsynaptic activity for accurate storage of temporal information.     

Neuromuscular synaptic transmission in Limulus polyphemus--I. Actions of aspartate, glutamate and the natural transmitter

Aspartate and glutamate were examined as excitatory transmitter candidates for the tibia flexor muscle of the chelicerate arthropod, Limulus polyphemus. Bath application of aspartate or glutamate caused dose-dependent depolarizations of Limulus muscle fibers and contractions of the whole muscle. Glutamate was about 10 times more potent than aspartate. Aspartate and glutamate depolarizations were associated with a conductance increase in muscle fibers, although aspartate depolarizations were dependent on external sodium, while glutamate depolarizations persisted in the absence of sodium. Although the Limulus excitatory postsynaptic potential (epsp) was associated with a conductance increase the ionic basis of the epsp could not be determined. If, however, the Limulus epsp, like other arthropod epsps, is sodium-dependent then the sodium-dependence of the aspartate depolarization is consistent with the action of the natural excitatory transmitter. The sodium-independence of glutamate action, however, is not consistent with generally accepted models of arthropod neuromuscular transmitter action. The rank order of potency for amino acid agonists indicates that the Limulus neuromuscular junction is pharmacologically very similar to other arthropod junctions which are well-accepted to be glutamatergic. Pentobarbital reversibly attenuated the amplitudes of the epsp and aspartate and glutamate depolarizations, and it was found to be the only useful antagonist in Limulus.     

The parameters of the stochastic leaky integrate-and-fire neuronal model

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data. Action Editor: Nicolas Brunel     

A homeostatic model of neuronal firing governed by feedback signals from the extracellular matrix

Molecules of the extracellular matrix (ECM) can modulate the efficacy of synaptic transmission and neuronal excitability. These mechanisms are crucial for the homeostatic regulation of neuronal firing over extended timescales. In this study, we introduce a simple mathematical model of neuronal spiking balanced by the influence of the ECM. We consider a neuron receiving random synaptic input in the form of Poisson spike trains and the ECM, which is modeled by a phenomenological variable involved in two feedback mechanisms. One feedback mechanism scales the values of the input synaptic conductance to compensate for changes in firing rate. The second feedback accounts for slow fluctuations of the excitation threshold and depends on the ECM concentration. We show that the ECM-mediated feedback acts as a robust mechanism to provide a homeostatic adjustment of the average firing rate. Interestingly, the activation of feedback mechanisms may lead to a bistability in which two different stable levels of average firing rates can coexist in a spiking network. We discuss the mechanisms of the bistability and how they may be related to memory function.     

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.     

Analytical and Simulation Results for Stochastic Fitzhugh-Nagumo Neurons and Neural Networks

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations—the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs withsuperimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.     

Random currents through nerve membranes

The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Green's function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits “1/f ^3/2 noise”. Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.     

Complementary responses to mean and variance modulations in the perfect integrate-and-fire model

In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance modulations are complementary, adding exactly to one. Since changes in the rate of Poisson distributed inputs lead to proportional changes in the mean and variance, these results imply that an ensemble of PIF neurons transmits a perfect replica of the time-varying input rate for Poisson distributed input. A more general argument shows that this property holds for any signal leading to proportional changes in the mean and variance of the input current.     

Neuron's firing time

The interspike interval distribution of neuronal firing is analyzed by a model that assumes unit effect EPSP's lasting an exponential length of time. The model allows a general interarrival distribution; this contrasts with the numerous models requiring Poisson arrivals. The Laplace transform of the time to firing, modelled as the first passage time to a fixed arbitrary threshold level, is found. Comparisons are made for exponential and regular interarrivals using the first two moments of the time to firing. Surprisingly, the mean and variance of the time to reach any threshold level greater than one is greater for regular arrivals for any ratio of mean interarrival intervals to mean EPSP duration greater than 0.6.     

Degradation of DNA in haemophilus influenzae cells after X-ray irradiation. II. Comparison with theoretical models.

Differential equations, describing time development of local sensitivity and that of ensemble average of firing, are obtained from a digital equation in a neural network. Theoretical structure giving the differential equation of the first order is found to be such that a slowly varying part of stimulus affects the variation of the averaged firing and its relaxation time non-linearly. High frequency part of stimulus with variation of about 1kHz is taken into account by being included in “effective threshold”, which is shown to contribute transmission probability between cells in division form. Effective threshold is also used to take into account rising phase of post synaptic potential firstly in theoretical neurology. The importance of effective threshold generated in a network with apt connections of inhibitory synapses is discussed as a future problem of information processing in a brain.     

Variability and randomness in stationary neuronal activity

The patterns of neuronal activity can be different even if the mean firing rate is fixed. Investigating the variability of the firing may not be sufficient and we suggest to take into account the notion of randomness. The randomness is related to the entropy of the firing, which is bounded from above by the entropy of the Poisson process (given the mean interspike interval). Thus, we propose the Kullback-Leibler distance with respect to the Poisson process as a measure of randomness in a stationary neuronal activity. Under the condition of equal mean values the KL distance does not depend on the time scale and therefore can be compared to the coefficient of variation employed to measure the variability. Furthermore, this measure can be extended to account for correlated neuronal firing. Finally, we analyze the variability and randomness for three common ISI distributions in detail: gamma, lognormal and inverse Gaussian.     

A Theoretical Analysis of Neuronal Variability

A simple neuronal model is assumed in which, after a refractory period, excitatory and inhibitory exponentially decaying inputs of constant size occur at random intervals and sum until a threshold is reached. The distribution of time intervals between successive neuronal firings (interresponse time histogram), the firing rate as a function of input frequency, the variability in the time course of depolarization from trial to trial, and the strength-duration curve are derived for this model. The predictions are compared with data from the literature and good qualitative agreement is found. All parameters are experimentally measurable and a direct test of the theory is possible with present techniques. The assumptions of the model are relaxed and the effects of such experimentally found phenomena as relative refractory and supernormal periods, adaptation, potentiation, and rhythmic slow potentials are discussed. Implications for gross behavior studies are considered briefly.     

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein–Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein–Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.     

Nonlinear systems analysis of computer models of repetitive firing

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.