Interpretation of the Repetitive Firing of Nerve Cells

Eccentric cells of Limulus respond with repetitive firing to sustained depolarizing currents. Following stimulation with a step of current, latency is shorter than first interval and later intervals increase progressively. A shock of intensity twice threshold can evoke firing 25 msec. after an impulse. But in the same cell, a current step twice rheobase evokes a second impulse more than 50 msec. after the first, and current intensity must be raised to over five times rheobase to obtain a first interval of about 25 msec. Repetitive firing was evoked by means of trains of shocks. With stimuli of moderate intensity, firing was evoked by only some of the shocks and intervals between successive impulses increased with time. This is ascribed to accumulation of refractoriness with successive impulses. Higher frequencies of firing are obtained with shocks of intensity n x threshold than with constant currents of intensity n x rheobase. It is concluded that prolonged currents depress the processes leading to excitation and that (in the cells studied) repetitive firing is controlled both by the after-effects of firing (refractoriness) and by the depressant effects of sustained stimuli (accommodation). Development of subthreshold "graded activity" is an important process leading to excitation of eccentric cells, but is not the principal factor determining frequency of firing in response to constant currents.     

Firing statistics of inhibitory neuron with delayed feedback. I. Output ISI probability density

Activity of inhibitory neuron with delayed feedback is considered in the framework of point stochastic processes. The neuron receives excitatory input impulses from a Poisson stream, and inhibitory impulses from the feedback line with a delay. We investigate here, how does the presence of inhibitory feedback affect the output firing statistics. Using binding neuron (BN) as a model, we derive analytically the exact expressions for the output interspike intervals (ISI) probability density, mean output ISI and coefficient of variation as functions of model's parameters for the case of threshold 2. Using the leaky integrate-and-fire (LIF) model, as well as the BN model with higher thresholds, these statistical quantities are found numerically. In contrast to the previously studied situation of no feedback, the ISI probability densities found here both for BN and LIF neuron become bimodal and have discontinuity of jump type. Nevertheless, the presence of inhibitory delayed feedback was not found to affect substantially the output ISI coefficient of variation. The ISI coefficient of variation found ranges between 0.5 and 1. It is concluded that introduction of delayed inhibitory feedback can radically change neuronal output firing statistics. This statistics is as well distinct from what was found previously (Vidybida and Kravchuk, 2009) by a similar method for excitatory neuron with delayed feedback.     

Firing rate of a retinal neuron are not predictable from interspike interval statistics.

The intervals between successive action potentials (impulses, or "spikes") produced the maintained firing of a neuron (ISIs) are often treated as if they were independent on each other; that is, an impulse train is considered as a stationary renewal process. If this is so, the variability of the mean rate of firing impulses in a sequence of temporal windows should be predictable from the distribution of ISIs. This was found not to be the case for the maintained firing of retinal ganglion cells in goldfish. Although some evident nonstationarity sometimes resulted in greater variability of the observed rate distributions than those predicted (for relatively long temporal windows), as a general rule the observed rate distributions were considerable less dispersed than would be predicted by sampling of the ISI distributions. This was taken as evidence of long-term serial dependency between successive ISIs; however, two standard test for dependency (autocorrelations and serial correlograms failed to to reveal structure of sufficiently long duration to account for the effect noted.     

Stability of delayed memristive neural networks with time-varying impulses

This paper addresses the stability problem on the memristive neural networks with time-varying impulses. Based on the memristor theory and neural network theory, the model of the memristor-based neural network is established. Different from the most publications on memristive networks with fixed-time impulse effects, we consider the case of time-varying impulses. Both the destabilizing and stabilizing impulses exist in the model simultaneously. Through controlling the time intervals of the stabilizing and destabilizing impulses, we ensure the effect of the impulses is stabilizing. Several sufficient conditions for the globally exponentially stability of memristive neural networks with time-varying impulses are proposed. The simulation results demonstrate the effectiveness of the theoretical results.     

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.     

Effects of afterhyperpolarization on neuronal firing.

Stein's model for a neuron is studied. This model is modified to take into account the effects of afterhyperpolarization on the neuronal firing. The relative refractory phase, following the absolute one, is modelled by a time-increasing amplitude of postsynaptic potentials and it is also incorporated into the model. Besides the simulation of the model, some theoretical results and approximation methods are derived. Afterhyperpolarization tends to preserve the linearity of the frequency transfer characteristic and it has a limited effect on the moments of the interspike intervals in general. The main effects are seen at high firing rates and in the removal of short intervals in the interspike interval histogram.     

Relationship between applicability of current-based synapses and uniformity of firing patterns

The purpose of this paper is to identify situations in neural network modeling where current-based synapses are applicable. The applicability of current-based synapse model for studying post-transient behavior of neural networks is discussed in terms of average synaptic current strength induced by per spike during one firing cycle of a neuron (or briefly per spike synaptic current strength). It was found that current-based synapse models are applicable in both situations where both the interspike intervals of the neurons and the distribution of firing times of the neurons are uniform, and where the firing of all neurons is synchronized. If neither the interspike intervals nor the distribution of firing times of the neurons is uniform or the reversal potential is between the rest and threshold potentials, current-based synapse models may be oversimplified.     

On the stationary state of a network of inhibitory spiking neurons

The background activity of a cortical neural network is modeled by a homogeneous integrate-and-fire network with unreliable inhibitory synapses. For the case of fast synapses, numerical and analytical calculations show that the network relaxes into a stationary state of high attention. The majority of the neurons has a membrane potential just below the threshold; as a consequence the network can react immediately – on the time scale of synaptic transmission- on external pulses. The neurons fire with a low rate and with a broad distribution of interspike intervals. Firing events of the total network are correlated over short time periods. The firing rate increases linearly with external stimuli. In the limit of infinitely large networks, the synaptic noise decreases to zero. Nevertheless, the distribution of interspike intervals remains broad. Action Editor: Misha Tsodyks     

神经放电起步点对电场刺激反应的“临界敏感”现象

在大鼠坐骨神经慢性压迫模型的放电起步点上,记录单纤维放电的峰峰间期(ISIs)序列。在无钙条件下,ISIs序列进入加周期分岔过程后,通过调定灌流液乙二醇双四乙酸(Ethylene Glycol—bis(β—aminoethyl Ether)N,N,N’,N’-Tetracetic Acid,EGTA,一种钙离子螯合剂)的浓度,使。ISIs序列分别稳定于远离分岔点的周期阶段(称周期阶段)或邻近分岔点的阶段(称临界阶段),分析电场刺激反应与分岔动力学状态的关系。实验观察到,相同强度的电场刺激可使周期阶段和临界阶段的放电频率增加,但后者的增加幅度比前者显著,并伴有放电模式的转化。在周期阶段,随电场刺激强度增大,放电频率近似线性增加,放电模式不变;在临界阶段,当电场刺激达到一定强度时,放电频率增加的斜率显著增大,此时,放电模式也发生转化。结果提示邻近分岔点的临界阶段对电场刺激的反应较周期阶段敏感,称之为“临界敏感”现象。     

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.     

Effect of the topology and delayed interactions in neuronal networks synchronization

As important as the intrinsic properties of an individual nervous cell stands the network of neurons in which it is embedded and by virtue of which it acquires great part of its responsiveness and functionality. In this study we have explored how the topological properties and conduction delays of several classes of neural networks affect the capacity of their constituent cells to establish well-defined temporal relations among firing of their action potentials. This ability of a population of neurons to produce and maintain a millisecond-precise coordinated firing (either evoked by external stimuli or internally generated) is central to neural codes exploiting precise spike timing for the representation and communication of information. Our results, based on extensive simulations of conductance-based type of neurons in an oscillatory regime, indicate that only certain topologies of networks allow for a coordinated firing at a local and long-range scale simultaneously. Besides network architecture, axonal conduction delays are also observed to be another important factor in the generation of coherent spiking. We report that such communication latencies not only set the phase difference between the oscillatory activity of remote neural populations but determine whether the interconnected cells can set in any coherent firing at all. In this context, we have also investigated how the balance between the network synchronizing effects and the dispersive drift caused by inhomogeneities in natural firing frequencies across neurons is resolved. Finally, we show that the observed roles of conduction delays and frequency dispersion are not particular to canonical networks but experimentally measured anatomical networks such as the macaque cortical network can display the same type of behavior.     

Synchronous Clusters in a Noisy Inhibitory Neural Network

We study the stability and information encoding capacity of synchronized states in a neuronal network model that represents part of thalamic circuitry. Our model neurons have a Hodgkin-Huxley-type low-threshold calcium channel, display postinhibitory rebound, and are connected via GABAergic inhibitory synapses.We find that there is a threshold in synaptic strength, _c, below which there are no stable spiking network states. Above threshold the stable spiking state is a cluster state, where different groups of neurons fire consecutively, and each neuron fires with the same cluster each time. Weak noise destabilizes this state, but stronger noise drives the system into a different, self-organized, stochastically synchronized state. Neuronal firing is still organized in clusters, but individual neurons can hop from cluster to cluster. Noise can actually induce and sustain such a state below the threshold of synaptic strength. We do find a qualitative difference in the firing patterns between small (10 neurons) and large (1000 neurons) networks.We determine the information content of the spike trains in terms of two separate contributions: the spike-time jitter around cluster firing times, and the hopping from cluster to cluster. We quantify the information loss due to temporally correlated interspike intervals. Recent experiments on the locust olfactory system and striatal neurons suggest that the nervous system may actually use these two channels to encode separate and unique information.     

A model of the mechanisms of formation of unit activity

Dependence of the temporal structure of the spike discharge of a neuron in a weakly interacting network on the characteristics of excitatory and inhibitory input flows and on cell parameters was analyzed by a mathematical model. The intensity of communication between individual neurons corresponded to the intensity of synaptic communication between real spinal neurons. The temporal course of trace and accommodation processes in the model was similar to that of these processes in real spinal neurons. Connection of inhibitory inputs and an increase in the intensity of their influences were shown to be equivalent to a decrease in the intensity of excitatory input flows. Changes in cell parameters had a significant effect on the spike discharge only in the case of weak input influences (the ratio of the amplitude of the combined ESP evoked by the input spike train to the threshold value of membrane potential at rest was about 1.2:1.0 to 1.4:1.0). An increase in the input flow intensity led to considerable reorganization of the firing pattern: Mean values of interspike intervals and their fluctuations were reduced, histograms of interspike intervals became more symmetrical, and periodic waves appeared on the autocorrelation histograms. It is concluded on the basis of these results and of data in the literature that the main factor determining reorganization of the temporal structure of unit activity in a network of weakly interacting cells is the intensity of the input flow.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 12, No. 2, pp. 199–207, March–April, 1980.     

Deviation of spontaneous firing in frog medullar auditory units from the renewal point process

Statistical characteristics of spontaneous activity (distribution of interspike intervals, hazard function, autocorrelation function, autocorrelation function for the process with mixed intervals, and interdependence between adjoining intervals) were analyzed for 123 neurons of the frog medullar dorsal nucleus (homolog of the mammalian cochlear nucleus). In the majority of cells, this activity was distinct from the Poisson process, and firing periodicity was noticed in some cases. In addition, deviations of the spontaneous activity from the renewal process were usually observed. Weak yet reliable positive interspike-interval correlation was typical of most neurons; however, a negative correlation between short adjoining intervals was recorded for some units. These data suggest the effects of memory in the activity of single neurons of the auditory system.     

Interactions of glutamate and dopamine in a computational model of the striatum

A network model of simplified striatal principal neurons with mutual inhibition was used to investigate possible interactions between cortical glutamatergic and nigral dopaminergic afferents in the neostriatum. Glutamatergic and dopaminergic inputs were represented by an excitatory synaptic conductance and a slow membrane potassium conductance, respectively. Neuronal activity in the model was characterized by episodes of increased action potential firing rates of variable duration and frequency. Autocorrelation histograms constructed from the action potential activity of striatal model neurons showed that reducing peak excitatory conductance had the effect of increasing interspike intervals. On the other hand, the maximum value of the dopamine-sensitive potassium conductance was inversely related to the duration of firing episodes and the maximal firing rates. A smaller potassium conductance restored normal firing rates in the most active neurons at the expense of a larger proportion of neurons showing reduced activity. Thus, a homogeneous network with mutual inhibition can produce equally complex dynamics as have been proposed to occur in a striatal network with two neuron populations that are oppositely regulated by dopamine. Even without mutual inhibition it appears that increased dopamine concentrations could partially compensate for the effects of reduced glutamatergic input in individual neurons.     

Afferent Stochastic Modulation of Crayfish Caudal Photoreceptor Units

When all roots to the sixth ganglion of the crayfish are cut, the caudal photoreceptor unit (PRU) fires at regular intervals. With an intact preparation, stimulation of caudal tactile hairs has predominantly inhibitory effects on the PRU: short bursts of afferent impulses, produced by momentary mechanical stimulation of tactile hairs, have (a) occasional immediate excitatory effect on the PRU, (b) prolonged inhibitory effect. The mean firing rate of the afferented and deafferented PRUs reacts similarly to a step increase in light, but the same unit fires faster after deafferentation. In the dark, deafferented units often fire paired or multiple pulses; the interval between pulses in a pair is similar to the short mode in afferented histograms. A fiber-optic probe of the caudal ganglion demonstrates the approximate location of the photosensitive element.     

Effect doublet impulse sequences on the transient and steady responses in the computer-simulated nerve cell

The effects of doublet impulse sequences of the excitatory axon on the output response as firing probability (pr.) in the computer-simulated nerve cell were examined. A simple model was formulated to simulate the nerve cell, including the property that the resetting potential is influenced by the final membrane potential in the previous stage before firing. The relationship between input sequences with alternating long and short interval at the same mean rate and the transient and steady responses of the nerve cell was investigated. In this simulation, three summarized results were obtained: i) The responses were very sensitive to changing small size of excitatory post-synaptic potential (EPSP), especially in the firing stage of the transient state. ii) In the transient state, the size of characteristic area of responses was depending upon the size of absolute refractory period (ARP). The rise for shorter intervals was faster than that for longer intervals, agreeing well with part of the experimental results from the crayfish claw opener muscles. The transient responses were almost finished before the fifth firing. iii) In the steady state, the doublet impulse sequences usually produced the minimum response or valley-like response at which the doublet interval T _dwas 20 and/or 25 ms. These effects related to the characteristic areas in the transient responses.     

Time and space distribution of coincident impulses in closed neural circuits in the sensorimotor cortex of the rabbit (the rhythmical defensive dominanta)

In the course of analysis of the conjugate unit activity of simultaneously recorded neurons in the sensorimotor cortex of rabbits, 22 closed neural circuits consisting of 3 or 4 neurons were considered. In the model of the defensive dominanta, 1-3 weeks after imposing rhythmic (2 s) activity to a rabbit, the distribution of coincident impulses was analyzed in real time. It was found out that the events when the coincident impulses of neural pairs were generated with two-second intervals could be shifted in time and space over a closed circuit of neurons in one direction. Two-second intervals between the coincident impulses of the neighboring pairs could be conjugate, i.e. the end of one interval in one pair coincided with the beginning of a two-second interval in the next pair. Conjugate intervals of the neighboring neural pairs could promote a pass-through of the information on the stimulus properties over the closed neuronal circuit, thus completing a full cycle. The longest passes-through lasted from 10 and 12 s. Also, more intricate variants of the information transfer were revealed. Thus, not only passes-through of the two- second intervals between the neuronal pairs were observed, but also, coincident impulses repeatedly occurred with this interval in some of the pairs of the circuits. The longest transitions lasted 16 and 22 s.     

Alternating predictive and short-term memory modes of entorhinal grid cells

Several lines of evidence indicate that the entorhinal cortex has memory functions, but such functions have not been previously found in grid cells, a cell type that provides major input to the hippocampus. We examined the firing of grid cells as rats crossed (runs) through grid cell vertices. We found that on some runs, firing tended to occur mostly inbound as the rat approached a vertex center while on other runs firing occurred mainly outbound. These results suggest that cells have a predictive mode (inbound firing) in which they represent a position ahead of the animal and a short term memory (STM) mode (outbound firing) in which they represent positions just passed through. Analysis of cell pairs showed that when vertex crossings were less than 1 second apart, the two cells tended to have the same mode. This indicates that modes are a network property. The tendency to have the same mode disappeared if crossings were separated by 2-3 seconds, suggesting that modes alternate on the time scale of seconds. There was a small but statistically significant behavioral correlate of modes: velocity was slightly less in the STM mode. Both modes were organized by theta and gamma oscillations. The results suggest that the dual requirement for hippocampal storage and recall is met by rapidly alternating modes appropriate for predicting the future and storing the recent past.