Threshold model for clustered firing of neurons

Two probabilistic threshold models for burst activity of cortical neurons are proposed. In model I every input impulse increases the summed effect of previous input impulses by one unit. The decay of the summed effect takes place in discrete steps of one unit. A response occurs on arrival of an input impulse, when a threshold value is attained.Although after a response the summed effect is not reset to zero, it cannot exceed the threshold either. The distribution of intervals can be resolved in two components, one for long and one for short intervals. In model II intervals of the short component are terminated by a multiple response instead of one response.     

Properties of repetitive firing in cat motor cortical neurons to membrane depolarizing currents

Acute experiments were carried out on immobilized cats under superficial pentobarbital (20 mg/kg) anesthesia to investigate the parameters of the rhythmic discharge of neurons in the motor cortex in the area of representation of the forelimb, evoked by passage of steps of depolarizing current through an intracellular microelectrode. The steady-state repetitive firing rate was found to be a linear function of the strength of the current passing through the membrane; no secondary range was discovered. The slope of the "discharge frequency — current" function (f/I=k) was 18±10.7 spikes/sec/nA. The regression line between the slope of the "discharge frequency-current" function and the input resistance (R_in) drawn by the method of least squares had the form k=0.68, R_in=–11.3. Two types of curve of adaptation of the discharge frequency to the stimulating current were found: exponential and undamped oscillations. The curve of latent period of the first action potential in the rhythmic discharge and the length of the first interspike interval as a function of current strength was shown to be a hyperbola, but with different scales along the abscissa. The connection between the properties of the dendritic tree and the parameters of the rhythmic discharge of cortical neurons is discussed.M. V. Lomonosov Moscow State University. Translated from Neirofiziologiya, Vol. 8, No. 5, pp. 476–482, September–October, 1976.     

Ion conductances related to development of repetitive firing in mouse retinal ganglion neurons in situ

ABSTRACT: In the retina, the ability to encode graded depolarizations into spike trains of variable frequency appears to be a specific property of retinal ganglion neurons (RGNs). To deduce the developmental changes in ion conductances underlying the transition from single to repetitive firing, patch-clamp recordings were performed in the isolated mouse retina between embryonic day 15 (E15) and postnatal day 5 (P5). Immature neurons of the E15 retina were selected according to their capacity to generate voltage-activated Na+ currents (I(Na)(v)). Identification of P5 RGNs was based on retrograde labeling, visualization of the axon, or the amplitude of I(Na)(v). At E15, half of the cells were excitable but none of them generated more than one spike. At P5, all cells were excitable and a majority discharged in tonic fashion. Ion conductances subserving maintenance of repetitive discharge were identified at P5 by exposure to low extracellular Ca2+, Cd2+, and charybdotoxin, all of which suppressed repetitive discharge. omega-Conotoxin GVIA and nifedipine had no effect. We compared passive membrane properties and a variety of voltage-activated ion channels at E15 and P5. It was found that the density of high voltage-activated (HVA) Ca2+ currents increased in parallel with the development of repetitive firing, while the density of Ni2+-sensitive low voltage-activated (LVA) Ca2+ currents decreased. Changes in density and activation kinetics of tetrodotoxin-sensitive Na+ currents paralleled changes in firing thresholds and size of action potentials, but seemed to be unrelated to maintenance of repetitive firing. Densities of A-type K+ currents and delayed rectifier currents did not change. The results suggest that HVA Ca2+ channels, and among them a toxin-resistant subtype, are specifically engaged in activation of Ca2+-sensitive K+ conductance and thereby account for frequency coding in postnatal RGNs.     

Ion channel density and threshold dynamics of repetitive firing in a cortical neuron model

Modifying the density and distribution of ion channels in a neuron (by natural up- and down-regulation, by pharmacological intervention or by spontaneous mutations) changes its activity pattern. In the present investigation, we analyze how the impulse patterns are regulated by the density of voltage-gated channels in a model neuron, based on voltage clamp measurements of hippocampal interneurons. At least three distinct oscillatory patterns, associated with three distinct regions in the Na-K channel density plane, were found. A stability analysis showed that the different regions are characterized by saddle-node, double-orbit, and Hopf bifurcation threshold dynamics, respectively. Single strongly graded action potentials occur in an area outside the oscillatory regions, but less graded action potentials occur together with repetitive firing over a considerable range of channel densities. The presently found relationship between channel densities and oscillatory behavior may be relevance for understanding principal spiking patterns of cortical neurons (regular firing and fast spiking). It may also be of relevance for understanding the action of pharmacological compounds on brain oscillatory activity.     

Cesium: postjunctional repetitive firing in sympathetic ganglion

Bullfrog ganglion cells in Cs⁺-Ringer's solution developed postjunctional repetitive spike responses to single preganglionic stimuli but not to single antidromic or direct stimuli. This action of Cs⁺ is equivalent to that of the neostigmine-like drugs and is apparently generated by primary action on presynaptic nerve terminals. Alterations of K⁺ and Na⁺ currents in the nerve terminal membrane could be the underlying mechanism.     

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.     

Excitability of single firing human motoneurones to single and repetitive stimulation (experiment and model)

The activity of single motoneurones of m. flexor carpi ulnaris (FCU) was investigated by recording their motor unit (MU) action potentials during weak and moderate voluntary muscle contractions. The MU firing rate range was 4.5-15 imp/s. The excitability of motoneurones was tested with a number of single stimuli eliciting a monosynaptic H-reflex of low amplitude. Two different indices were defined which relate to motoneuronal excitability: the response index--the ratio of the number of responses of a motoneurone to the total number of stimuli, and the response time--the time after the last background MU discharge at which motoneurone is ready to respond to the excitatory volley. Both the response index and the response time were determined for single motoneurones at different levels of background activity. In the lower range of firing rates, the response index for all motoneurones decreased when increasing the firing rate, but it remained constant in the higher rate range. This kind of response seems to be a typical motoneuronal response to the stimulation with single stimuli. The data on the response time were used to study the excitability of the same single motoneurones to computer simulated repetitive stimulation (stimulation rate 40-100 imp/s). In this case, the excitability of each motoneurone was determined as an increment of its firing rate in response to the stimulation. For the lower firing rate range, the excitability for all motoneurones also decreased when the firing rates increased whereas a variety of slopes was obtained in the higher rate range.(ABSTRACT TRUNCATED AT 250 WORDS)     

A model for refractoriness accumulation and secondary range firing in spinal motoneurones

A model is presented in which the potassium conductance (G _K ) changes responsible for the afterhyperpolarization (AHP) in motoneurones are postulated to follow a kinetics coherent with Hodgkin-Huxley membrane model. Such G _K kinetics, which operates as a bistable system switched from one state to the other by the action potential, can be expressed by simple analytical expressions if the spike is approximated to a rectangular pulse. Accumulation of G _K by repetitive activation of the model accurately describes the different features of AHP summation in motoneurones; moreover, this accumulation process enables a model for repetitive firing of motoneurones to display secondary range firing at steady state.     

After-effects of repetitive stimulation in spinal neurons

Spontaneous activity of interneurons before and after repetitive stimulation at 0.1–0.5/sec was recorded in acute experiments on spinal cats and kittens. Using the dynamic selective correlation method a search was made for areas of spontaneous activity with the same distribution of action potentials in time as in the averaged evoked response to a single stimulus. In the case of some neurons portions of the background which correlate reliably in structure with the evoked response repeated at an interval equal to or a multiple of the interval of stimulation. Reproduction of the rhythm of stimulation in the spontaneous activity is intensified with an increase in the total duration of preceding stimulation with the same input and shows positive correlation with the degree of posttetanic potentiation. The facts obtained are evidence of prolonged after-processes in spinal neurons.Institute of Experimental Medicine, Academy of Medical Sciences of the USSR, Leningrad. Translated from Neirofiziologiya, Vol. 5, No. 3, pp. 272–280, May–June, 1973.     

Modelling natural burst firing in nigral dopamine neurons

The natural burst firing observed in vivo in mesolimbic dopamine neurons is of great significance regarding these neurons' involvement in response to sensory stimuli associated with primary reward. The cellular mechanisms underlying a natural burst have been experimentally characterized previously and hypothesized to be caused by a calcium-sensitive inactivation of a potassium channel. We present a mathematical model of a mesolimbic neuron that demonstrates how such a mechanism can produce realistic bursting patterns, but only when combined with an appropriately timed membrane depolarization from an external source.     

A model for the firing pattern of a paced nerve cell

When a nerve cell is paced by another cell or by external stimuli, its firing pattern is generally not regular. In a given stimulus and frequency interval the cell responds only on a fraction of the stimuli, and this fraction is not necessarily a nice simple fraction as 2, ;, etc. but may be any fraction less than one, why the firing pattern becomes accordingly irregular.The paper demonstrates the rules for these irregularities and supplies a mathematical tool to analyse the firing pattern. In the analysis only a cell with the so-called impulse dependent type of adaptation is considered, but similar analyses may be done with other cell types.The analysis is restricted to a case where a resting cell is stimulated by repetitively applied stimuli of constant strength and repetition frequency. Cases with more than one input, spontaneously firing cells, and transient phenomena are not included in the analysis.     

A basic biophysical model for bursting neurons

Presented here is a basic biophysical cell model for bursting, an extension of our previous model (Av-Ron et al. 1991) for excitability and oscillations. By changing a limited set of model parameters, one can describe different patterns of bursting behavior in terms of the burst cycle, the durations of oscillation and quiescence, and firing frequency.     

Modeling repetitive firing and bursting in a small unmyelinated nerve fiber.

The Hodgkin-Huxley equations, originally developed to describe the electrical events in the squid giant axon, have been modified to simulate the ionic and electrical events in a small unmyelinated nerve fiber. The modified equations incorporate an electrogenic sodium-potassium pump, finite intra-axonal volume, a periaxonal space, a calcium current, and calcium-dependent potassium conductance (GKCa). The model shows that adaptation can occur in two ways: increased Na-K pump activity because of periaxonal K accumulation or intra-axonal Na accumulation; or from an increase in (GKCa) caused by calcium accumulating within the axon. Bursting is an extension of adaptation and occurs when the sensitivity of the Na-K pump or (GKCa) to changes in ionic concentration is increased.