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.     

Accuracy of neuronal interspike times calculated from a diffusion approximation

The theory of neuronal firing in Stein's model is outlined as well as the corresponding theory for a diffusion approximation which has the same first two infinitesimal moments. The diffusion approximation is derived from the discontinuous model in the limit of large input frequencies and small postsynaptic potential amplitudes. A comparison of the calculated mean interspike intervals is made for various values of the threshold for firing and various input frequencies. The diffusion approximation can underestimate the interspike interval by up to 100% or severely overestimate it, depending on the input frequencies and the threshold. A general relation between the predictions of the two models is deduced.     

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.     

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.     

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.     

Modelling Feedback Excitation,Pacemaker Properties and Sensory Switching of Electrically Coupled Brainstem Neurons Controlling Rhythmic Activity

What cellular and network properties allow reliable neuronal rhythm generation or firing that can be started and stopped by brief synaptic inputs? We investigate rhythmic activity in an electrically-coupled population of brainstem neurons driving swimming locomotion in young frog tadpoles, and how activity is switched on and off by brief sensory stimulation. We build a computational model of 30 electrically-coupled conditional pacemaker neurons on one side of the tadpole hindbrain and spinal cord. Based on experimental estimates for neuron properties, population sizes, synapse strengths and connections, we show that: long-lasting, mutual, glutamatergic excitation between the neurons allows the network to sustain rhythmic pacemaker firing at swimming frequencies following brief synaptic excitation; activity persists but rhythm breaks down without electrical coupling; NMDA voltage-dependency doubles the range of synaptic feedback strengths generating sustained rhythm. The network can be switched on and off at short latency by brief synaptic excitation and inhibition. We demonstrate that a population of generic Hodgkin-Huxley type neurons coupled by glutamatergic excitatory feedback can generate sustained asynchronous firing switched on and off synaptically. We conclude that networks of neurons with NMDAR mediated feedback excitation can generate self-sustained activity following brief synaptic excitation. The frequency of activity is limited by the kinetics of the neuron membrane channels and can be stopped by brief inhibitory input. Network activity can be rhythmic at lower frequencies if the neurons are electrically coupled. Our key finding is that excitatory synaptic feedback within a population of neurons can produce switchable, stable, sustained firing without synaptic inhibition.     

The Role of Inhibition in a Computational Model of an Auditory Cortical Neuron during the Encoding of Temporal Information

In auditory cortex, temporal information within a sound is represented by two complementary neural codes: a temporal representation based on stimulus-locked firing and a rate representation, where discharge rate co-varies with the timing between acoustic events but lacks a stimulus-synchronized response. Using a computational neuronal model, we find that stimulus-locked responses are generated when sound-evoked excitation is combined with strong, delayed inhibition. In contrast to this, a non-synchronized rate representation is generated when the net excitation evoked by the sound is weak, which occurs when excitation is coincident and balanced with inhibition. Using single-unit recordings from awake marmosets (Callithrix jacchus), we validate several model predictions, including differences in the temporal fidelity, discharge rates and temporal dynamics of stimulus-evoked responses between neurons with rate and temporal representations. Together these data suggest that feedforward inhibition provides a parsimonious explanation of the neural coding dichotomy observed in auditory cortex.     

Analysis of a stochastic neuronal model with excitatory inputs and state-dependent effects

We propose a stochastic model for the firing activity of a neuronal unit. It includes the decay effect of the membrane potential in absence of stimuli, and the occurrence of time-varying excitatory inputs governed by a Poisson process. The sample-paths of the membrane potential are piecewise exponentially decaying curves with jumps of random amplitudes occurring at the input times. An analysis of the probability distributions of the membrane potential and of the firing time is performed. In the special case of time-homogeneous stimuli the firing density is obtained in closed form, together with its mean and variance.     

Synchronization and sensitivity enhancement of the Hodgkin-Huxley neurons due to inhibitory inputs

Recent experimental results imply that inhibitory postsynaptic potentials can play a functional role in realizing synchronization of neuronal firing in the brain. In order to examine the relation between inhibition and synchronous firing of neurons theoretically, we analyze possible effects of synchronization and sensitivity enhancement caused by inhibitory inputs to neurons with a biologically realistic model of the Hodgkin-Huxley equations. The result shows that, after an inhibitory spike, the firing probability of a single postsynaptic neuron exposed to random excitatory background activity oscillates with time. The oscillation of the firing probability can be related to synchronous firing of neurons receiving an inhibitory spike simultaneously. Further, we show that when an inhibitory spike input precedes an excitatory spike input, the presence of such preceding inhibition raises the firing probability peak of the neuron after the excitatory input. The result indicates that an inhibitory spike input can enhance the sensitivity of the postsynaptic neuron to the following excitatory spike input. Two neural network models based on these effects on postsynaptic neurons caused by inhibitory inputs are proposed to demonstrate possible mechanisms of detecting particular spatiotemporal spike patterns. Received: 15 April 1999 /Accepted in revised form: 25 November 1999     

An analysis of the reliability phenomenon in the FitzHugh-Nagumo model

The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes—the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.     

Shunting inhibition modulates neuronal gain during synaptic excitation

Neuronal gain control is important for processing information in the brain. Shunting inhibition is not thought to control gain since it shifts input-output relationships during tonic excitation rather than changing their slope. Here we show that tonic inhibition reduces the gain and shifts the offset of cerebellar granule cell input-output relationships during frequency-dependent excitation with synaptic conductance waveforms. Shunting inhibition scales subthreshold voltage, increasing the excitation frequency required to attain a particular firing rate. This reduces gain because frequency-dependent increases in input variability, which couple mean subthreshold voltage to firing rate, boost voltage fluctuations during inhibition. Moreover, synaptic time course and the number of inputs also influence gain changes by setting excitation variability. Our results suggest that shunting inhibition can multiplicatively scale rate-coded information in neurons with high-variability synaptic inputs.     

Analysis of the activity of single neurons in stochastic settings

This paper presents a new way of modeling the activity of single neurons in stochastic settings. It incorporates in a natural way many physiological mechanisms not usually found in stochastic models, such as spatial integration, non-linear membrane characteristics and non-linear interactions between excitation and inhibition. The model is based on the fact that most of the neuronal inputs have a finite lifetime. Thus, the stochastic input can be modeled as a simple finite markov chain, and the membrane potential becomes a function of the state of this chain. Firing occurs at states whose membrane potential is above threshold. The main mathematical results of the model are: (i) the input-output firing rate curve is convex at low firing rates and is saturated at high firing rates, and (ii) at low firing rates, firing usually occurs when there is synchronous convergence of many excitatory events.     

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.     

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.     

A Simulation Study to Examine the Effect of Common Motoneuron Inputs on Correlated Patterns of Motor Unit Discharge

The influence of common oscillatory inputs to the motoneuron pool on correlated patterns of motor unit discharge was examined using model simulations. Motor unit synchronization, in-phase fluctuations in mean firing rates known as ‘common drive’, and the coefficient of variation of the muscle force were examined as the frequency and amplitude of common oscillatory inputs to the motoneuron pool were varied. The amount of synchronization, the peak correlation between mean firing rates and the coefficient of variation of the force varied with both the frequency and amplitude of the common input signal. Values for ‘common drive’ and the force coefficient of variation were highest for oscillatory inputs at frequencies less than 5 Hz, while synchronization reached a maximum when the frequency of the common input was close to the average motor unit firing rate. The frequency of the common input signal for which the highest levels of synchronization were observed increased as motoneuron firing rates increased in response to higher target force levels. The simulation results suggest that common low-frequency oscillations in motor unit firing rates and short-term synchronization result from oscillatory activity in different bands of the frequency spectrum of shared motoneuron inputs. The results also indicate that the amount of synchronization between motor unit discharges depends not only on the amplitude of the shared input signal, but also on its frequency in relation to the present firing rates of the individual motor units.     

Profound alterations in the intrinsic excitability of cerebellar Purkinje neurons following neurotoxin 3-acetylpyridine (3-AP)-induced ataxia in rat: new insights into the role of small conductance K+ channels

Alterations in the intrinsic properties of Purkinje cells (PCs) may contribute to the abnormal motor performance observed in ataxic rats. To investigate whether such changes in the intrinsic neuronal excitability could be attributed to the role of Ca(2+)-activated K(+) channels (K(Ca)), whole cell current clamp recordings were made from PCs in cerebellar slices of control and ataxic rats. 3-AP induced profound alterations in the intrinsic properties of PCs, as evidenced by a significant increase in both the membrane input resistance and the initial discharge frequency, along with the disruption of the firing regularity. In control PCs, the blockade of small conductance K(Ca) channels by UCL1684 resulted in a significant increase in the membrane input resistance, action potential (AP) half-width, time to peak of the AP and initial discharge frequency. SK channel blockade also significantly decreased the neuronal discharge regularity, the peak amplitude of the AP, the amplitude of the afterhyperpolarization and the spike frequency adaptation ratio. In contrast, in ataxic rats, both the firing regularity and the initial firing frequency were significantly increased by the blockade of SK channels. In conclusion, ataxia may arise from alterations in the functional contribution of SK channels, to the intrinsic properties of PCs.     

Postinhibitory unit response of crustacean stretch receptors after direct and recurrent inhibition

The postinhibitory response of a slowly adapting neuron was investigated in experiments on an isolated preparation of crustacean stretch receptor and abdominal nerve chain. The structural features of this preparation are such that this response can be regarded as the response of the postsynaptic membrane to synaptic inhibition and not the action of synaptic excitation. IPSPs arise in the slowly adapting neuron in response to stimulation of the abdominal nerve chain (direct inhibition) or to excitation of the neuron itself (recurrent inhibition). The postinhibitory response consists of the development of action potentials or an increase in their amplitude and frequency. The magnitude of the response is determined by the duration of the inhibition and the state of the neuron membrane. The postinhibitory response was strongest when IPSPs were superposed on cathodal depression. IPSPs and an intracellular hyperpolarizing current evoke similar postinhibitory responses. Repetitive excitation of an inhibitory neuron may result in the appearance of a regular spike discharge from a previously inactive neuron through the mechanism of the postinhibitory response. Activation of a chain of recurrent inhibition increases the duration of the postinhibitory response evoked by direct inhibition or by a hyperpolarizing current. The existence of a chain of recurrent inhibition prevents the cessation of firing by a neuron during increasing cathodal depression. A mechanism of postinhibitory rebound lies at the basis of this phenomenon.     

Firing statistics of inhibitory neuron with delayed feedback. II: Non-Markovian behavior

The instantaneous state of a neural network consists of both the degree of excitation of each neuron the network is composed of and positions of impulses in communication lines between the neurons. In neurophysiological experiments, the neuronal firing moments are registered, but not the state of communication lines. But future spiking moments depend essentially on the past positions of impulses in the lines. This suggests, that the sequence of intervals between firing moments (inter-spike intervals, ISIs) in the network could be non-Markovian.     

Single unit activity in the guinea-pig cochlear nucleus during sleep and wakefulness.

The effects of waking and sleep on the response properties of auditory units in the ventral cochlear nucleus (CN) were explored by using extracellular recordings in chronic guinea-pigs. Significant increases and decreases in firing rate were detected in two neuronal groups, a) the "sound-responding" and b) the "spontaneous" (units that do not show responses to any acoustic stimuli controlled by the experimenter). The "spontaneous" may be considered as belonging to the auditory system because the corresponding units showed a suppression of their discharge when the receptor was destroyed. The auditory CN units were characterized by their PSTH in response to tones at their characteristic frequency and also by the changes in firing rate and probability of discharge evaluated during periods of waking, slow wave and paradoxical sleep. The CNS performs functions dependent on sensory inputs during wakefulness and sleep phases. By studying the auditory input at the level of the ventral CN with constant sound stimuli, it was shown that, in addition to the firing rate shifts, some units presented changes in the temporal probability of discharge, implying central actions on the corresponding neurons. The mean latency of the responses, however, did not show significant changes throughout the sleep-waking cycle. The auditory efferent pathways are postulated to modulate the auditory input at CN level during different animal states. The probability of firing and the changes in the temporal pattern, as shown by the PSTH, are thus dependent on both the auditory input and the functional brain state related to the sleep-waking cycle.