Further consideration on pattern separability in a random neural net with inhibitory connections

A two-layer random neural net with inhibitory connections composing of threshold elements has been regarded as a model of the cerebellar cortex. Many properties of pattern separation with the model have been disclosed through consideration on the degree of pattern separation. However, we have not shown yet that the degree of pattern separation is given by some different functions which are decided by the relation between the firing rates of input patterns. The present study is intended to reveal that the functions of the degree of pattern separation are synthesized with some different partial functions, and they are differently given on the relation between the firing rates of input patterns. Simultaneously, it is proved that the number of the functions also depend on the number of connections between two layers in the model. We also disclose the properties of the degree of pattern separation, and give some suggestions on the sizes of the firing rates of mossy fibers and granule cells under the knowledge about them.     

Pattern separability and the effect of the number of connections in a random neural net with inhibitory connections

It has been claimed that pattern separation in cerebellar cortex plays an important role in controlling movements and balance for vertebrates. A number of the neural models for cerebellar cortex have been proposed and their pattern separability has been analyzed. These results, however, only explain a part of pattern separability in random neural nets. The present paper is intended to study an extended theory of pattern separability in a new model with inhibitory connections. In addition to this, the effect of the number of connections on pattern separability is cleared up. It is also shown that the signal from the inhibitory connections has crucial importance for pattern separability.1977–1978 Exchange Visitor, on leave from the Department of Information Processing Engineering, Technical College, Yamaguchi University, Yamaguchi University     

Simulation of spontaneous discharge and short-term adaptation in acoustic nerve fibers

A model for firing of the auditory nerve fibres was carried out on a general purpose digital computer. In the model a noise with small correlation time and with assigned standard deviation (when modeling a spontaneous discharge) or a sum of a noise and a determinated signal (when modeling an elicited discharge) is compared with incremental threshold. When the threshold is exceeded a spike occurs and the threshold is increased. The threshold qualitative properties and quantitative values were chosen in a way to provide the most reliable patterns of spontaneous discharge, according to the literature data obtained from the cat's auditory nerve. When stimulated by tone-bursts the model reveals intrinsic ability of mimicking the phenomena of discharge rate short-term adaptation. Thus according to our model the short-term adaptation is entirely due to the properties of the incremental threshold.     

Plausible function of Golgi cells in the cerebellar cortex

The function of Golgi cells in the cerebellar cortex is quantitatively examined in consideration of the nonlinear input-output characteristics and convergence and divergence numbers of cells. It is strongly suggested that the two signal paths to Golgi cells have different function. The feed-forward path will have the same function as assumed in the previous theories of the cerebellar cortex, that is, to keep the firing rate of granule cells approximately constant over considerable variation in the firing rate of mossy fibers. The feedback path will, on the other hand, have a new function which has not been assumed in the previous theories. The function is to cause oscillation of the firing rate of granule cells for stationary mossy fiber inputs. The assumption of the new function enables us to explain cerebellar function to keep stationary posture.     

Delayed reverberation through time windows as a key to cerebellar function

We present a functional model of the cerebellum comprising cerebellar cortex, inferior olive, deep cerebellar nuclei, and brain stem nuclei. The discerning feature of the model being time coding, we consistently describe the system in terms of postsynaptic potentials, synchronous action potentials, and propagation delays. We show by means of detailed single-neuron modeling that (i) Golgi cells can fulfill a gating task in that they form short and well-defined time windows within which granule cells can reach firing threshold, thus organizing neuronal activity in discrete `time slices', and that (ii) rebound firing in cerebellar nuclei cells is a robust mechanism leading to a delayed reverberation of Purkinje cell activity through cerebellar-reticular projections back to the cerebellar cortex. Computer simulations of the whole cerebellar network consisting of several thousand neurons reveal that reverberation in conjunction with long-term plasticity at the parallel fiber-Purkinje cell synapses enables the system to learn, store, and recall spatio-temporal patterns of neuronal activity. Climbing fiber spikes act both as a synchronization and as a teacher signal, not as an error signal. They are due to intrinsic oscillatory properties of inferior olivary neurons and to delayed reverberation within the network. In addition to clear experimental predictions the present theory sheds new light on a number of experimental observation such as the synchronicity of climbing fiber spikes and provides a novel explanation of how the cerebellum solves timing tasks on a time scale of several hundreds of milliseconds. Received: 23 July 1999 / Accepted in revised form: 31 August 1999     

A fully coupled transient excited state model for the sodium channel

Summary The axon membrane is simulated by standard Hodgkin-Huxley leakage and potassium channels plus a coupled transient excited state kinetic scheme for the sodium channel. This scheme for the sodium channel is as proposed previously by the author. Simultations are presented showing the form of the action potential, threshold behavior, accommodation, and repetitive firing. It is seen that the form of the individual action potential, its all-or-none nature, and its refractory period are well simulated by this model, as they are by the standard Hodgkin-Huxley model. However, the model differs markedly from the Hodgkin-Huxley model with respect to repetitive firing and accommodation to stimulating currents of slowly rising intensity, in ways that are anomn to be related to those features of the sodium inactivation which are anomalous to the H-H model. The tendency for repetitive firing is highly dependent on that parameter which primarily determintes the existence of the inactivation shift in voltage clamp experiments, in such a way that the more pronounced the inactivation shift, the less the tendency for repetitive firing,. The tendency for accommodation is highly dependent on that parameter which primarily determines the “τ_c − τ_h” separation, in such a way that the greater the separation the greater the tendency for the membrane to accommodate without firing action potentials to a slowly rising current.     

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     

A general diffusion model for analyzing the efficacy of synaptic input to threshold neurons

We describe a general diffusion model for analyzing the efficacy of individual synaptic inputs to threshold neurons. A formal expression is obtained for the system propagator which, when given an arbitrary initial state for the cell, yields the conditional probability distribution for the state at all later times. The propagator for a cell with a finite threshold is written as a series expansion, such that each term in the series depends only on the infinite threshold propagator, which in the diffusion limit reduces to a Gaussian form. This procedure admits a graphical representation in terms of an infinite sequence of diagrams. To connect the theory to experiment, we construct an analytical expression for the primary correlation kernel (PCK) which profiles the change in the instantaneous firing rate produced by a single postsynaptic potential (PSP). Explicit solutions are obtained in the diffusion limit to first order in perturbation theory. Our approximate expression resembles the PCK obtained by computer simulation, with the accuracy depending strongly on the mode of firing. The theory is most accurate when the synaptic input drives the membrane potential to a mean level more than one standard deviation below the firing threshold, making such cells highly sensitive to synchronous synaptic input.     

Cerebellar Cortex Granular Layer Interneurons in the Macaque Monkey Are Functionally Driven by Mossy Fiber Pathways through Net Excitation or Inhibition

The granular layer is the input layer of the cerebellar cortex. It receives information through mossy fibers, which contact local granular layer interneurons (GLIs) and granular layer output neurons (granule cells). GLIs provide one of the first signal processing stages in the cerebellar cortex by exciting or inhibiting granule cells. Despite the importance of this early processing stage for later cerebellar computations, the responses of GLIs and the functional connections of mossy fibers with GLIs in awake animals are poorly understood. Here, we recorded GLIs and mossy fibers in the macaque ventral-paraflocculus (VPFL) during oculomotor tasks, providing the first full inventory of GLI responses in the VPFL of awake primates. We found that while mossy fiber responses are characterized by a linear monotonic relationship between firing rate and eye position, GLIs show complex response profiles characterized by “eye position fields” and single or double directional tunings. For the majority of GLIs, prominent features of their responses can be explained by assuming that a single GLI receives inputs from mossy fibers with similar or opposite directional preferences, and that these mossy fiber inputs influence GLI discharge through net excitatory or inhibitory pathways. Importantly, GLIs receiving mossy fiber inputs through these putative excitatory and inhibitory pathways show different firing properties, suggesting that they indeed correspond to two distinct classes of interneurons. We propose a new interpretation of the information flow through the cerebellar cortex granular layer, in which mossy fiber input patterns drive the responses of GLIs not only through excitatory but also through net inhibitory pathways, and that excited and inhibited GLIs can be identified based on their responses and their intrinsic properties.     

Effect of Structure on Function in Model Nerve Nets

A theoretical analysis has been made on the effect of the pattern of interneuronal connectivity in model nerve nets on the activity of these nets. Two types of nets have been investigated: one in which the likelihood of a connection between a given neuron and any other element in the net is given by a Poisson probability distribution, and a second type in which the pattern of interconnection follows a Gaussian distribution. An analytical treatment is presented of the equations for noiseless nets in these two conditions. The principal result is that nets with Poisson connectivity law are activated by extraneous firing of a single neuron and continue in spontaneous activity indefinitely. On the other hand, similar nets in which the connections are, however, distributed according to a normal connectivity law, exhibit a definite threshold and produce spontaneous activity only subsequent to extraneous activation of a substantial fraction of the population. Moreover, spontaneous activity in Gaussian nets, but not in Poisson nets, becomes extinguished if the number of active neurons falls below the critical threshold. Some neuroanatomical implications are discussed which suggest that the pyramidal system of the cerebral cortex and other neuronal systems histologically characterized by large numbers of synapses per neuron may incorporate a Gaussian connectivity law, whereas a Poisson law may be characteristic of these cortical layers and nuclei primarily containing granule cells.     

BK Channels Control Cerebellar Purkinje and Golgi Cell Rhythmicity In Vivo

Calcium signaling plays a central role in normal CNS functioning and dysfunction. As cerebellar Purkinje cells express the major regulatory elements of calcium control and represent the sole integrative output of the cerebellar cortex, changes in neural activity- and calcium-mediated membrane properties of these cells are expected to provide important insights into both intrinsic and network physiology of the cerebellum. We studied the electrophysiological behavior of Purkinje cells in genetically engineered alert mice that do not express BK calcium-activated potassium channels and in wild-type mice with pharmacological BK inactivation. We confirmed BK expression in Purkinje cells and also demonstrated it in Golgi cells. We demonstrated that either genetic or pharmacological BK inactivation leads to ataxia and to the emergence of a beta oscillatory field potential in the cerebellar cortex. This oscillation is correlated with enhanced rhythmicity and synchronicity of both Purkinje and Golgi cells. We hypothesize that the temporal coding modification of the spike firing of both Purkinje and Golgi cells leads to the pharmacologically or genetically induced ataxia.     

Transformation of Adaptation and Gain Rescaling along the Whisker Sensory Pathway

Neurons in all sensory systems have a remarkable ability to adapt their sensitivity to the statistical structure of the sensory signals to which they are tuned. In the barrel cortex, firing rate adapts to the variance of a whisker stimulus and neuronal sensitivity (gain) adjusts in inverse proportion to the stimulus standard deviation. To determine how adaptation might be transformed across the ascending lemniscal pathway, we measured the responses of single units in the first and last subcortical stages, the trigeminal ganglion (TRG) and ventral posterior medial thalamic nucleus (VPM), to controlled whisker stimulation in urethane-anesthetized rats. We probed adaptation using a filtered white noise stimulus that switched between low- and high-variance epochs. We found that the firing rate of both TRG and VPM neurons adapted to stimulus variance. By fitting the responses of each unit to a Linear-Nonlinear-Poisson model, we tested whether adaptation changed feature selectivity and/or sensitivity. We found that, whereas feature selectivity was unaffected by stimulus variance, units often exhibited a marked change in sensitivity. The extent of these sensitivity changes increased systematically along the pathway from TRG to barrel cortex. However, there was marked variability across units, especially in VPM. In sum, in the whisker system, the adaptation properties of subcortical neurons are surprisingly diverse. The significance of this diversity may be that it contributes to a rich population representation of whisker dynamics.     

Cortical Network Models of Firing Rates in the Resting and Active States Predict BOLD Responses

Measurements of blood oxygenation level dependent (BOLD) signals have produced some surprising observations. One is that their amplitude is proportional to the entire activity in a region of interest and not just the fluctuations in this activity. Another is that during sleep and anesthesia the average BOLD correlations between regions of interest decline as the activity declines. Mechanistic explanations of these phenomena are described here using a cortical network model consisting of modules with excitatory and inhibitory neurons, taken as regions of cortical interest, each receiving excitatory inputs from outside the network, taken as subcortical driving inputs in addition to extrinsic (intermodular) connections, such as provided by associational fibers. The model shows that the standard deviation of the firing rate is proportional to the mean frequency of the firing when the extrinsic connections are decreased, so that the mean BOLD signal is proportional to both as is observed experimentally. The model also shows that if these extrinsic connections are decreased or the frequency of firing reaching the network from the subcortical driving inputs is decreased, or both decline, there is a decrease in the mean firing rate in the modules accompanied by decreases in the mean BOLD correlations between the modules, consistent with the observed changes during NREM sleep and under anesthesia. Finally, the model explains why a transient increase in the BOLD signal in a cortical area, due to a transient subcortical input, gives rises to responses throughout the cortex as observed, with these responses mediated by the extrinsic (intermodular) connections.     

The Sodium-Potassium Pump Controls the Intrinsic Firing of the Cerebellar Purkinje Neuron

In vitro, cerebellar Purkinje cells can intrinsically fire action potentials in a repeating trimodal or bimodal pattern. The trimodal pattern consists of tonic spiking, bursting, and quiescence. The bimodal pattern consists of tonic spiking and quiescence. It is unclear how these firing patterns are generated and what determines which firing pattern is selected. We have constructed a realistic biophysical Purkinje cell model that can replicate these patterns. In this model, Na⁺/K⁺ pump activity sets the Purkinje cell''s operating mode. From rat cerebellar slices we present Purkinje whole cell recordings in the presence of ouabain, which irreversibly blocks the Na⁺/K⁺ pump. The model can replicate these recordings. We propose that Na⁺/K⁺ pump activity controls the intrinsic firing mode of cerbellar Purkinje cells.     

Spontaneous bioelectric activity of cultured purkinje cells during exposure to glutamate,glycine, and strychnine

The addition of glutamate to the bathing medium increased the average firing rate of cerebellar rat Purkinje cells in vitro. At concentrations lower than 10^?6 M, there was no deviation from controls in the firing pattern or rate that was detectable. At 10^?3 M glutamate, the amplitude of the action potentials was gradually decreased until all activity was abolished. The action of glutamate was rapid in onset and reversible. Glycine produced sustained depression of firing at concentrations higher than 10^?3 M. This inhibition was strychnine-insensitive and considered nonspecific. Strychnine, on the other hand, exerted an excitatory influence on Purkinje cells when applied at low concentrations (10^?8 to 10^?6 M). The firing became more irregular and complex discharges appeared. Higher concentrations of strychnine (>10^?5 M) inhibited the spontaneous activity. The effect of strychnine was partly reversible. The data suggest that low concentrations of strychnine lower the threshold for inputs at excitatory as well as inhibitory synapses.     

Dynamic Statistics of Crayfish Caudal Photoreceptors

Crayfish caudal photoreceptor units were monitored during transient and steady-state responses to light stimuli (step on, step off). A statistical analysis of interpulse interval distributions during quasi-stationary time periods was carried out. Firing statistics during transient conditions were superposable with statistics under whatever steady stimulation produced the same firing rate, indicating that mean firing rate is a sufficient statistic. Distributions encountered formed a continuum of possible shapes. Considerable variation in shape was found with temperature and also among species, with Orconectes clarkii tending to fire more regularly than Orconectes virilis. Some properties of O. virilis statistics are described, including a linear relation between mean and standard deviation, and a tendency for intervals to be nonindependent. The data are considered as constraints on closed form models of the photoreceptor nerve pulse generator.     

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.     

Predicting the Responses of Repetitively Firing Neurons to Current Noise

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.     

The Output Signal of Purkinje Cells of the Cerebellum and Circadian Rhythmicity

Measurement of clock gene expression has recently provided evidence that the cerebellum, like the master clock in the SCN, contains a circadian oscillator. The cerebellar oscillator is involved in anticipation of mealtime and possibly resides in Purkinje cells. However, the rhythmic gene expression is likely transduced into a circadian cerebellar output signal to exert an effective control of neuronal brain circuits that are responsible for feeding behavior. Using electrophysiological recordings from acute and organotypic cerebellar slices, we tested the hypothesis whether Purkinje cells transmit a circadian modulated signal to their targets in the brain. Extracellular recordings from brain slices revealed the typical discharge pattern previously described in vivo in single cell recordings showing basically a tonic or a trimodal-like firing pattern. However, in acute sagittal cerebellar slices the average spike rate of randomly selected Purkinje cells did not exhibit significant circadian variations, irrespective of their specific firing pattern. Also, frequency and amplitude of spontaneous inhibitory postsynaptic currents and the amplitude of GABA- and glutamate-evoked currents did not vary with circadian time. Long-term recordings using multielectrode arrays (MEA) allowed to monitor neuronal activity at multiple sites in organotypic cerebellar slices for several days to weeks. With this recording technique we observed oscillations of the firing rate of cerebellar neurons, presumably of Purkinje cells, with a period of about 24 hours which were stable for periods up to three days. The daily renewal of culture medium could induce circadian oscillations of the firing rate of Purkinje cells, a feature that is compatible with the behavior of slave oscillators. However, from the present results it appears that the circadian expression of cerebellar clock genes exerts only a weak influence on the electrical output of cerebellar neurons.     

Determination of the fraction of active inputs required by a neuron to fire

What fraction of the inputs to a neuron in the primary visual cortex (V1) need to be active for that neuron to reach its firing threshold? The paper describes a numerical method for estimating the selectivity of visual neurons, in terms of the required fraction of active excitatory inputs, from standard data produced by intracellular electro-physiological recordings. The method also provides an estimate of the relative strength of the feedforward inhibition in a push-pull model of the inputs to V1 simple cells. The method is tested on two V1 cells described in Carandini and Ferster [Carandini, M., Ferster, D., 2000. Membrane potential and firing rate in cat primary visual cortex. J. Neurosci. 20, 470-484]. The results indicate that the maximum strength of feedforward inhibition is around 30% of the maximum strength of feedforward excitation. The two V1 neurons investigated fire if more than around 40% of their excitatory LGN inputs are active.