Firing-rate models capture essential response dynamics of LGN relay cells

Firing-rate models provide a practical tool for studying signal processing in the early visual system, permitting more thorough mathematical analysis than spike-based models. We show here that essential response properties of relay cells in the lateral geniculate nucleus (LGN) can be captured by surprisingly simple firing-rate models consisting of a low-pass filter and a nonlinear activation function. The starting point for our analysis are two spiking neuron models based on experimental data: a spike-response model fitted to data from macaque (Carandini et al. J. Vis., 20(14), 1–2011, 2007), and a model with conductance-based synapses and afterhyperpolarizing currents fitted to data from cat (Casti et al. J. Comput. Neurosci., 24(2), 235–252, 2008). We obtained the nonlinear activation function by stimulating the model neurons with stationary stochastic spike trains, while we characterized the linear filter by fitting a low-pass filter to responses to sinusoidally modulated stochastic spike trains. To account for the non-Poisson nature of retinal spike trains, we performed all analyses with spike trains with higher-order gamma statistics in addition to Poissonian spike trains. Interestingly, the properties of the low-pass filter depend only on the average input rate, but not on the modulation depth of sinusoidally modulated input. Thus, the response properties of our model are fully specified by just three parameters (low-frequency gain, cutoff frequency, and delay) for a given mean input rate and input regularity. This simple firing-rate model reproduces the response of spiking neurons to a step in input rate very well for Poissonian as well as for non-Poissonian input. We also found that the cutoff frequencies, and thus the filter time constants, of the rate-based model are unrelated to the membrane time constants of the underlying spiking models, in agreement with similar observations for simpler models.     

Mathematical models of human paralyzed muscle after long-term training

Spinal cord injury (SCI) results in major musculoskeletal adaptations, including muscle atrophy, faster contractile properties, increased fatigability, and bone loss. The use of functional electrical stimulation (FES) provides a method to prevent paralyzed muscle adaptations in order to sustain force-generating capacity. Mathematical muscle models may be able to predict optimal activation strategies during FES, however muscle properties further adapt with long-term training. The purpose of this study was to compare the accuracy of three muscle models, one linear and two nonlinear, for predicting paralyzed soleus muscle force after exposure to long-term FES training. Further, we contrasted the findings between the trained and untrained limbs. The three models' parameters were best fit to a single force train in the trained soleus muscle (N=4). Nine additional force trains (test trains) were predicted for each subject using the developed models. Model errors between predicted and experimental force trains were determined, including specific muscle force properties. The mean overall error was greatest for the linear model (15.8%) and least for the nonlinear Hill Huxley type model (7.8%). No significant error differences were observed between the trained versus untrained limbs, although model parameter values were significantly altered with training. This study confirmed that nonlinear models most accurately predict both trained and untrained paralyzed muscle force properties. Moreover, the optimized model parameter values were responsive to the relative physiological state of the paralyzed muscle (trained versus untrained). These findings are relevant for the design and control of neuro-prosthetic devices for those with SCI.     

Transient bifurcations in neural error correction

This paper presents an investigation into the responses of neurons to errors in presynaptic spike trains. Errors are viewed, in nonlinear dynamical terms, as brief-duration changes in stationary presynaptic spike trains which induce transient responses in the postsynaptic cell. As these are generally large-magnitude transients, linearized neural models are not helpful. Instead, the responses of a full, nonlinear physiological model of a neuron that includes the recognized living prototype of an inhibitory synapse are analyzed. More specifically, the transients are examined in the context of the stationary behaviors that precede and succeed each error. It is shown that one and two dimensional bifurcation diagrams can be constructed from the transient responses--that there are marked changes in the transient responses at points that correspond to bifurcations in the stationary responses, qualitative changes in transients on either side of bifurcations, and only quantitative changes in transients between bifurcations.     

Mathematical models for fatigue minimization during functional electrical stimulation

We previously reported the development of a force- and fatigue-model system that predicted accurately forces during repetitive fatiguing activation of human skeletal muscles using brief duration (six-pulse) stimulation trains. The model system was tested in the present study using force responses produced by longer duration stimulation trains, containing up to 50 pulses. Our results showed that our model successfully predicted the peak forces produced when the muscle was repetitively activated with stimulation trains of frequencies ranging from 20 to 40 Hz, train durations ranging from 0.5 to 1 s, and varied pulse patterns. The predicted peak forces throughout each protocol matched the experimental peak forces with r² values above 0.9 and predicted successfully the forces at the end of each protocol with <15% error for all protocols tested. The success of our model system further supports its potential use for the design of optimal stimulation patterns for individual users during functional electrical stimulation.     

Optimal modeling of an arterial system

The aortic blood flow is described by a set of nonlinear hyperbolic partial differential equations that account for mass and momentum conservation, and nonlinear models for the mechanical properties of the artery. Identification is used for determining the wave speed, arterial taper, and cross section: these parameters reflect the elastic characteristics of the aorta wall and control the pulsatile response. The differential equations were numerically integrated by the Lax-Wendroff scheme of Abarbanel and Goldberg [J. Comput. Phys. 10:1–21 (1972)] that avoids nonlinear oscillations. The Gauss-Newton technique was used for the parameter identification. By reference to reported elocity and pressure input-output pairs, a parameter vector is found such that the distance in the L₂ norm between the predicted outputs and the measured functions is minimal. Calculations of the velocity and pressure waves show excellent compatibility of the model with reported experimental data: starting from arbitrary parameter estimates, which yield grossly distorted waveforms, the error is typically reduced to 7–8%. Introduction of viscoelastic behaviour for the arterial wall in the form of a Volterra integral for the cross section does not lead to significant improvement. Numerical examples are presented which prove the convergence, accuracy, and stability of the algorithm. Emphasis is placed on the computational feasibility of the proposed system identification.     

Transmission of spike trains by nerve fibers after near-threshold stimulation

Spike trains in group A nerve fibers were studied in anesthetized cats in response to stimulation of the hind-limb nerves by random (Poisson) and regular sequences of stimuli. In response to above-threshold stimulation spike trains in the nerve fibers were shown to differ from the stimulating trains purely in the absence of intervals less than 1–1.5 msec in duration, as a result of the presence of a refractory period. With near-threshold stimulation with an average frequency of over 10 per second, spike trains differed significantly from the stimulating trains, as reflected in histograms of interspike intervals, the shape of the intensity function, and the magnitude of the coefficient of correlation for successive intervals. It is postulated that changes in the structure of the spike trains conveyed by a nerve fiber are attributable to the presence of after-activity.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 8, No. 1, pp. 91–98, January–February, 1976.     

Effects of applying electrical stimulation to the locus coeruleus on neuronal activity in the parietal association cortex

It was shown during experiments on cats undergoing surgery under ketamine-induced anesthesia and immobilized with myorelaxin that applying trains of stimuli to the locus coeruleus (LC) produces an effect on 79% of parietal cortex neurons. This manifests as inhibition lasting 300–700 msec or a 16–32% decline in the activity rate of neurons with background activity. Hyperpolarization of 5–7 mV lasting 120–500 msec preceded by a latency of 30–90 msec was noted in such neurons as well as "silent" cells during intracellular recording. Duration of the inhibitory pause in neuronal background activity induced by transcallosal stimulation (TCS) increased by 50–200 msec under the effects of conditioned stimuli applied to the LC. Duration of the IPSP triggered by TCS likewise increased (by 50–100 msec) under the effects of LC stimulation. It was concluded that the effects of stimulating the LC on neuronal activity in the parietal cortex may manifest either directly, as inhibition of background activity and hyperpolarization, or else as modulation of influences exerted by other neurotransmitters.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 22, No. 4, pp. 486–494, July–August, 1990.     

Investigation of a monosynaptic excitatory connection by means of a biomathematical model of neuronal interaction

A biomathematical model of neuronal interaction, including real mollusk neurons and mathematical models of functioning of deficient synaptic connections between these neurons and synaptic endings of other neurons, was created on the basis of a computer and an experimental arrangement for investigating molluscan ganglia. The effect of the properties of a monosynaptic excitatory connection of the statistical characteristics of spike trains of interacting pacemaker neurons was investigated.Institute of Higher Nervous Activity and Neurophysiology, Academy of Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 12, No. 4, pp. 413–420, July–August, 1980.     

Nonlinear characteristics of changes in active muscle length

The principle nonlinear characteristics of changes in the length of active (soleus, gastrocnemius, and plantaris) muscle resulting from controlled changes in external load were examined during acute experiments on anesthetized cats. Summation of successive muscle responses to repetitive phased changes in load was shown to be absent due to hysteresis effects; this does not satisfy the principles of superposition and leads to an important functional result: the muscle exerts a stabilizing effect on overall motor system dynamics, limiting unwanted shifts in joint angles during variation in external load. A relationship between the trajectory profile of change in muscle length and the lead-up to the movement arises due to muscle contraction hysteresis. Velocity at the initial stage of movement was always higher when the latter was preceded by motion in the same direction. The functional significance of the nonlinear properties of active muscle movement accompanying changing external load is discussed.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 20, No. 6, pp. 736–743, November–December, 1988.     

Characterization of spatial and temporal properties of monkey LGN Y-cells

LGN Y-cells in 3 anaesthetized (N₂O/O₂) and paralyzed rhesus monkeys were investigated with stimuli, intensity modulated by gaussian white noise, and with moving and counterphase modulated spatial sine wave gratings. The results support the model, postulated on the base of electrophysiological recordings in the retina of cat and mudpuppy, which consists of a linear centre and surround mechanism whose responses are modified in a frequency-selective multiplicative way by a nonlinear mechanism in the receptive field. This nonlinear mechanism is also held responsible for the second-order harmonic responses, which are the defining characteristic of Y-cells. The temporal and spatial characteristics of these mechanisms were determined. The responses obtained with the GWN stimulation and with modulated spatial sine wave gratings both indicate that the optimal temporal frequency of the linear mechanisms is near 7 Hz at 70 td and near 5 Hz for the nonlinear mechanism. The optimal spatial frequency for the linear mechanism is between 0.5–2 cycles/deg and between 6–12 cycles/deg for the nonlinear mechanism.     

A study of the response properties of retinula cells of flies using nonlinear identification theory

The functional properties of retinula cells of the fly Calliphora crythrocephala (wild type) have been determined through the application of nonlinear identification theory using white-gaussian stimulus functions. These results are also compared to similiar recordings of lamina cells. The accuracy of the resulting models is shown by the fact that those obtained from just 30 sec tests predict the actual total responses to the white-gaussian stimuli with a mean square error of about 5% and for a 2200 sec test the error is reduced to 2%. It has been shown that the second order kernels define all of the nonlinear properties and are the primary terms in the models for describing the variations in functional responses with illumination level, changing adaptation conditions and even variations in the conditions of the intracellular preparations.     

Prediction of lower critical solution temperature of N-isopropylacrylamide–acrylic acid copolymer by an artificial neural network model

In this paper, we have investigated the lower critical solution temperature (LCST) of N-isopropylacrylamide–acrylic acid (NIPAAm-AAc) copolymer as a function of chain-transfer agent/initiator mole ratio, acrylic acid content of copolymer, concentration, pH and ionic strength of aqueous copolymer solution. Aqueous solutions with the desired properties were prepared from previously purified polymers, synthesized at 65 °C by solution polymerization using ethanol. The effects of each parameter on the LCST were examined experimentally.In addition, an artificial neural network model that is able to predict the lower cretical solution temperature was develeped. The predictions from this model compare well against both training and test data sets with an average error less than 2.53%.Figure Cross plot of predicted and experimental LCST values for the testing data set.     

Neuronal interaction in the auditory cortex of alert cats

Spontaneous activity of neighboring auditory cortical neurons was derived by glass microelectrodes in chronic experiments on unanesthetized, unimmobilized cats, and the spike trains were subsequently analyzed by computer. Altogether 20 pairs of neurons were tested. The commonest type of interaction (50%) was found to be a common excitatory source, conjectured to be from specific auditory afferents. Interaction of the "common inhibitory source" (5%) and also complex forms of interaction were found. Interaction was absent in only 10% of cases. No direct inhibitory influence of neighboring neurons on one another was observed. The possible causes of absence of a direct inhibitory action are discussed. The most likely cause is absence of marked spontaneous activity in inhibitory auditory cortical neurons.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 16, No. 2, pp. 161–167, March–April, 1984.     

Responses of neurons of reticular and ventral anterior nuclei of the cat thalamus to afferent stimuli of different modalities

Responses of 146 spontaneously active neurons of the reticular nucleus (R) and of 98 neurons of the ventral anterior (VA) nucleus of the thalamus to electrical stimulation of the skin of the footpads, to flashes, and to clicks were studied in experiments on cats immobilized with D-tubocurarine or myorelaxin. Stimulation of the contralateral forelimb was the most effective: 24.9% of R neurons and 31.3% of VA neurons responded to this stimulation. A response to clicks was observed in only 4.4% of R neurons and 2.4% of VA neurons. Nearly all responding neurons did so by phasic (one spike or a group of spikes) or tonic excitation. Depression of spontaneous activity was observed only in response to electrical stimulation of the skin. Depending on the site of stimulation, it was observed in 2.6–4.3% of R neurons and 1.7–2.1% of VA neurons tested. The latent period of the phasic responses of most neurons was 6–64 msec to electrical stimulation of the contralateral forelimb, 11–43 msec in response to stimulation of the hindlimb on the same side, 10–60 msec to photic and 8–60 msec to acoustic stimulation. Depending on the character of stimulation, 75.1–95.6% of R neurons and 68.7–97.6% of VA cells did not respond at all to the stimuli used. Of the total number of cells tested against the whole range of stimuli, 25% of R neurons and 47% of VA neurons responded to stimulation of different limbs, whereas 16% of R neurons and 22% of VA cells responded to stimuli of different sensory modalities. The functional role of the convergence revealed in these experiments is to inhibit (or, less frequently, to facilitate) the response of a neuron to a testing stimulus during the 40–70 msec after conditioning stimulation.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 7, No. 6, pp. 563–571, November–December, 1975.     

Role of the midbrain periaqueductal gray matter in conditioned reflexes

Unit activity in the midbrain periaqueductal gray matter (PGM) during an instrumental placing reflex, its extinction, differentiation, and conditioned inhibition, was studied in chronic experiments on cats. Spike responses 1–2 sec in duration in 69 (36.7%) of 182 neurons preceded by 400–800 msec the beginning of conditioned-reflex and voluntary intertrial movements. These advanced responses appeared 200 msec before the corresponding advance responses of motor cortical neurons. Fifty-eight neurons (30.9%) responded directly to acoustic stimulation with a latent period of 10–50 msec for 2–6 sec, 19 neurons (10.1%) generated double responses, linked with both the acoustic stimulus and subsequent conditioned-reflex movement, and 42 neurons (22.3%) did not respond to acoustic stimulation, although individual neurons of this group changed the level of their spontaneous activity in response to repeated conditioned stimulation, and this change was maintained for some tens of minutes. Extinction, differentiation, and conditioned inhibition all abolished conditioned-reflex movements, but each type of internal inhibition was accompanied by its own characteristic changes in the firing pattern of PGM neurons. Functional independence of neurons of the first and second groups was demonstrated during extinction and recovery of the conditioned-reflex. The results indicate the important role of PGM not only in the mechanism of the conditioned reflex, but also in the development of its internal inhibition.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 16, No. 3, pp. 403–419, May–June, 1984.     

Unit responses of the pericruciate cortex to stimulation of the medial geniculate body

Responses of 239 neurons of the pericruciate cortex to stimulation of the medial geniculate body and pyramidal tract were investigated (189 extracellularly, 50 intracellularly) in cats anesthetized with thiopental and immobilized with D-tubocurarine. In response to stimulation of the medial geniculate body, the mean spontaneous firing rate of 63.6% of neurons in the pericruciate cortex increased by 10–25%, in 23.6% of neurons it decreased within the same limits, and mixed effects were observed in 5.5% of neurons. Phasic responses to single stimulation of the medial geniculate body were observed in 20% of neurons of the pericruciate cortex. Responses with a latent period of 0.3–1.0 msec (16%) were classed as antidromic, those with a latent period of 1.5–2.0 msec (20%) as orthodromic, monosynaptic, and those with a latent period of 2.5–4.0 msec or more (64%) as polysynaptic. With intracellular recording, excitatory responses of the EPSP, EPSP-AP, and AP type with latent periods of between 1.3 and 19.5 msec developed in 78.2% of cells. IPSPs, which were recorded in 21.8% of neurons, were usually found as components of mixed responses; primary IPSPs were found in only two cases. Monosynaptic connection of the medial geniculate body was shown to take place with neurons of the pericruciate cortex that did not belong to the pyramidal tract.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 11, No. 1, pp. 18–24, January–February, 1979.     

Functional properties distinguishing individual auditory cortical neurons

Responses of 246 auditory cortical neurons to paired and repetitive stimulation of geniculo-cortical fibers were studied in experiments on cats immobilized with tubocurarine. The refractory period (RP) varied from 1 to 200 msec in different neurons. For neurons excited antidromically it varied from 1 to 3 msec. Among neurons excited monosynaptically there were some with a short (1.3–6 msec), medium, (8–16 msec) or long (30–100 msec) refractory period. Most neurons excited polysynaptically had a RP of mean length. RPs 30–200 msec in length were due to inhibition arising in the neuron after conditioning stimulation. In some neurons, after a short (1.5–2.0 msec) initial period of refractoriness there was a temporary (for 2–3 msec) recovery of responsiveness, followed by another period of ineffectiveness of the testing stimulus lasting 30–100 msec. Barbiturates selectively inhibited long-latency unit responses in the auditory cortex and during their action the number of responding neurons with a mean RP decreased sharply. The results demonstrate functional heterogeneity of auditory cortical neurons responding to an incoming volley of afferent impulses.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 5, No. 3, pp. 236–245, May–June, 1973.     

Responses of limbic cortical neurons during instrumental conditioned reflexes in cats

Unit activity in cortical areas 24 and 32 was studied during conditioned placing reflex formation in cats. Neuronal responses in the limbic cortex of trained animals correlated with acoustic stimulation, the motor response, and also with the presentation of food reinforcement. In untrained animals 16% of neurons responded to acoustic stimulation. After training the number of neurons responding to sound in area 32 increased to 51.3%. Of the total number of neurons, 34.6% responded by initial excitation and 26.7% by inhibition of spike activity. The latent period of these responses was about 50 msec and their duration up to 200 msec. Similar but weaker responses were observed in area 24. Short-latency activation responses to conditioned and differential stimulation were similar in character. It is suggested that after training processes taking place in the limbic cortex may contribute to better perception of both conditioned and differential acoustic stimuli, irrespective of their functional significance.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 16, No. 2, pp. 201–208, March–April, 1984.