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.     

The effects of recurrent inhibition on the cross-correlated firing patterns of motoneurones (and their relation to signal transmission in the spinal cord-muscle channel)

Cross-correlation histograms (CCH) were computed for discharge sequences of pairs of motoneurones which were excited by sinusoidal muscle stretches. These CCH's were compared before and after opening of the recurrent inhibitory loop by Renshaw cell blocking agents. Periodic patterns in the CCH's indicative of specifically timed phase relations between discharges of different motoneurones were enhanced after Renshaw cell blockage. This was confirmed by power spectra computed for the CCH's. They contained power peaks about 50Hz which tended to increase after depression of recurrent inhibition. The correlation was thus due predominantly to line current interference which seemed to act as a common entrainment input at the spinal level. It is concluded that Renshaw cells de-correlate discharge patterns of different motoneurones of the same pool by injecting uncorrelated signals into them. This de-correlation is an important prerequisite for distortion suppression of signal transmission in a multi-channel system, like that of stretch reflex, and for its linearization.     

Excitability cycle of the Hoffmann reflex of the soleus in a constant pool of motoneurons. Changes in man as a function of age

The study of the recovery cycle of the H reflex of the soleus, in constant pool, was made on 88 normal men of different ages. The recurrent inhibition has been demonstrated in most of the adults and elderly subjects. In young children, there were powerful inhibitory mechanisms, among which the Renshaw inhibition can be isolated.     

The sensitivity of Renshaw cells to velocity of sinusoidal stretches of the triceps surae muscle.

1. The electrical activity of Renshaw cells monosynaptically excited by ventral root stimulation and disynaptically excited by electric stimulation of the group I afferents in the GS nerve has been recorded and their response to individual sinusoidal stretches of the deefferented GS muscle tested for different amplitudes and durations of the stimulus. 2. The experimental data indicate that the Rensahw cell responses are not only length dependent but also rate dependent. This finding indicates that the same Renshaw cells receive recurrent collaterals of both tonic and phasic motoneurons. 3. The observation that the discharge of Renshaw cells is particularly sensitive to the velocity of stretch suggests that the recurrent collaterals of large phasic motoneurons, which are recruited during high velocity stretches, exert a stronger excitatory action on Renshaw cells than do axon collaterals of the smaller tonic motoneurons, which are selectively stimulated during low velocity stretches.     

Static and dynamic input-output relations of the feline medial gastrocnemius motoneuron-muscle system subjected to recurrent inhibition: a model study

The physiological function of spinal recurrent inhibition is still a matter of debate because of the experimental difficulty or impossibility of observing recurrent inhibition at work in normally behaving animals. The purpose of this study was to investigate, by computer simulation, the role of recurrent inhibition in shaping the input-output (I/O) relationships between descending command signals (DCS) as inputs and motoneuron (MN) and Renshaw cell (RC) firing rates and muscle force as outputs. Changing the spatial (topographical) distribution of recurrent inhibition from nonhomogeneous (as in the standard model) to homogeneous did not alter the I/O relationships significantly, while changing the functional distribution related to MN types did. Altering the global gain of recurrent inhibition, as happens naturally in various motor acts, changes the slopes and positions (at high inputs) of the I/O relationships, making recurrent inhibition a suitable means of gain control. Coupling a decrease in recurrent inhibitory gain with an increase in DCS input, as could occur during slow dynamic contractions, would increase the MN and force gains during the act. Short dynamic ramp-and-hold DCS inputs generate MN firing patterns, to which recurrent inhibition contributes interspike-interval variability and damped oscillations, which are related to issues of tremor and its control.     

The effects of recurrent inhibitory feedback in shaping discharge patterns of motoneurones excited by phasic muscle stretches

The spinal motoneurone-Renshaw cell circuitry, which constitutes an intricate negative feedback system, was investigated with respect to its significance for the shaping of motoneurone firing patterns excited by strong phasic inputs. Discharge patterns of single motoneurones were compared before and after opening of the recurrent inhibitory pathways by Renshaw cell blocking agents. Serial correlograms computed from motoneurone interspike intervals which were modulated by sinusoidal muscle stretch replicated the input periodicity, but were not changed in any consistent manner after Renshaw cell blockage. Longterm regularity and periodicity of motoneurone firing, i.e. the overall response characteristics, do not appear to be significantly determined by Renshaw cells under these conditions. The modulation of motoneurone interspike intervals was assessed by computing power spectra for corresponding instantaneous frequencies. The harmonic contents (2^nd and 3^rd harmonic of the driving frequency) of these spectra tended to decrease after Renshaw cell depression. The distortion of signal transmission in a single motoneurone channel is thus stronger with, than without, the recurrent inhibitory feedback. The implication of these findings for signal transmission from the spinal cord to the muscle is discussed.     

Dynamic behaviour of α motoneurone sub-pools subjected to inhomogeneous Renshaw cell inhibition

The spinal α-motoneurone-Renshaw cell system was simulated by a meshed system of three principal negative feedback loops interconnected via “cross”-feedback pathways. Three types of α-motoneurone (MN): S-type, FR-type, and FF-type MNs, and their differing connections to and from Renshaw cells (RCs) were taken into account. The dynamic behaviour of RCs was taken from data provided by Cleveland and Ross (1977) and assumed to be given by a transfer function with one zero and two poles whose time constants τ_i depended on the overall amount of excitatory input to RCs. Also, the static gain of recurrent inhibition was taken to decrease with increasing excitatory input from α-MN axon collaterals (Cleveland et al., 1981) and to be depressed by spinally descending motor command signals. S-type MNs as well as F-type MNs were assumed to have high-pass characteristics though with slightly different cut-off frequencies. The closed-loop frequency responses of each sub-pool of MNs, S, FR, and FF, at three different levels of recruitment of these sub-pools, were calculated and shown to change significantly with recruitment level. These changes were essentially due to two reasons: firstly, to the general reduction of static gains within the recurrent inhibitory pathways with increasing motor output (recruitment), and secondly, to the increasing complexity of the whole network by recruitment of each new MN type. The particularly strong effect of the latter factor could easily be demonstrated by a comparison of the frequency responses of the MN types when these were, firstly, integrated into the network at their particular level of recruitment, and when they were, secondly, hypothetically assumed “isolated” from the remaining network, i.e., when subjected only to “self-inhibition”, the cross-inhibitory links to other MN types being cut. These results illustrate that the dynamic behaviour of α-MNs submitted to an inhomogeneously distributed recurrent and variable inhibition are not invariant, but depend upon the variable characteristics of a complex MN-RC network. This suggests that an important physiological function of recurrent inhibition via Renshaw cells, particularly of its inhomogeneous distribution, may be to adjust the dynamic MN sensitivity to the particular requirements prevailing at different motor output levels.     

A study and model of the role of the Renshaw cell in regulating the transient firing rate of the motoneuron

 This study sought to investigate the role of the Renshaw cell with respect to transient motoneuron firing. By studying the cat motoneuron and Renshaw cell, several low-order lumped parameter models were developed that simulate the known characteristics of the injected input current vs. firing rate. The neuron models in the Renshaw cell inhibition configuration were tuned to fit experimental data from cat motoneurons. Models included both linear versions and those with sigmoidal nonlinearities. Results of the simulation indicate that the motoneuron itself provides the adaptation seen in its firing rate and that the Renshaw cell’s role is primarily to fine-tune the motoneuron’s adaptation process. Received: 23 July 1993/Accepted in revised form: 9 February 1994     

Static input-output relations in the spinal recurrent inhibitory pathway

The static discharge rate of Renshaw cells (studied in deafferented, intercollicularly decerebrate cats) has a nonlinear dependence on the frequency of trains of stimulus impulses to -motor axons in the ventral root. This dependence is well described by a rectangular hyperbola that approaches saturation with increasing stimulus frequency. The tendency to saturate is independent of the number of motor axons exciting a Renshaw cell. On average, the stimulus frequency at which the discharge rate reaches half its saturation value lies between 10 and 15 Hz. The effect of Renshaw cell activity — measured as the antidromic inhibition of individual -motoneurons — reflects the form of the static frequency characteristics. An electric circuit analog of the Renshaw cell membrane is presented which serves to explain the qualitative features of the static input-output relations; the nonlinearity is the result of synapses with linear properties acting together at the cell membrane.Dedicated to Professor R. Granit, Stockholm, on the occasion of his 80th birthday     

Modeling different regimes of bioelectrical activity in the brain in normal subjects and in "increased readiness" in a network of neuron-like elements

Desynchronous (low voltage fast activity), synchronous (high voltage slow waves) as well as convulsive brain activities were stimulated by a computer model of neuronal population. Network excitatory and inhibitory elements possessed fundamental dynamic properties of real neurones. Being independent both of the excitability of elements and of external influence efficacy, synchronous (desynchronous) network activity resulted from the increase (decrease) of the average power of "neuronal" interconnections which imitated mutual and recurrent excitation and inhibition. The inhibition efficacy being reduced as compared with excitation, synchronization of elements became intensified. As a consequence, the rhythmic activity amplitude increased and the appearance of self-sustained oscillations simulating convulsive activity was facilitated. The probable mechanism of EEG activation by virtue of the reduction of mutual and recurrent excitation and inhibition efficacy as well as the significance of inhibitory mechanism deficiency for epileptogenesis are discussed.     

Excitability cycle of the soleus motor nucleus during synergistic conditioning in man

Electrical stimulation of medial or lateral gastrocnemius nerve modulates the H reflex of the soleus muscle in a well defined sequence of inhibitory and facilitatory phases the amplitude of which depends on the strength of the conditioning stimulus. This method of investigation of the soleus monosynaptic reflex pathway avoids the disadvantages of homonymous conditioning. Discussion of our results points to the roles of Ib and Renshaw cells inhibitions in the early phases of the recovery cycle, and to the roles of reafferents, disinhibition of Renshaw cells, presynaptic inhibition and muscarinic discharge of Renshaw cells for the late phases.     

The relative sensitivity of Renshaw cells to orthodromic group Ia volleys caused by static stretch and vibrations of extensor muscles.

1. Activity of Renshaw cells monosynaptically excited by ventral root stimulation and disynaptically excited by electric stimulation of the group Ia afferents in the gastrocnemius-soleus (GS) nerve, was recorded in precollicular decerebrate cats. The response of these units to prolonged vibration applied longitudinally to the deefferented GS muscle was then compared with that elicited by static stretch of the homonymous muscle, for comparable frequencies of discharge of the group Ia afferents. 2. Small-amplitude vibration of the GS muscle at 200/sec for one second produced a sudden increase in the discharge rate of Renshaw cells, which gradually decreased within the first 100 msec of vibration to reach steady albeit lower level than that obtained during the first part of vibration. The response of the Renshaw cells during the first 100 msec of vibration (phasic response) and that elicited during the last 500 msec of vibration (tonic response) were evaluated for different frequencies of sinusoidal stretch. The mean increase in the firing frequency per imp./sec in the Ia afferents was also calculated using the total one-second period. 3. The response of Renshaw cells to muscle vibration increased with the frequency of vibration and, over the value of 10/sec, appeared to be linearly related to the frequency of the input, at least up to the frequency of 150/sec. Since vibration was of sufficient amplitude to produce driving of all the primary endings of muscle spindles, the responses were expressed as mean increases in the discharge rate of Renshaw cells per average impulse/sec in the Ia afferents. The discharge of the Renshaw cell increased on the average by 2.90 and 1.08 imp./sec per each imp./sec in the Ia afferents during the phasic and the tonic component of the response respectively, while the response calculated during the whole period of vibration corresponded on the average to 1.45 imp./sec per each imp./sec in the Ia afferents. 4. The Renshaw cells tested above responded also with increasing frequencies of discharge to increasing levels of static extension of the GS muscle. In particular the discharge frequency of Renshaw cells was on the average linearly related to muscle extension, at least for values ranging from 0 to 8 mm. The mean increase in discharge rate as a function of the static extension corresponded on the average to 0.89 imp./sec/mm. Since the discharge rate of the primary endings of muscle spindles recorded from the deefferented GS muscle increased by 2.62 imp./sec/mm, it appears that the mean increase in the discharge rate of Renshaw cells as a function of static extension corresponded to 0.34 imp./sec per each imp./sec in the Ia afferents.     

Effects of spinal recurrent inhibition on motoneuron short-term synchronization

Spinal recurrent inhibition linking skeleto- motoneurons (α-MNs) via Renshaw cells (RCs) has been variously proposed to increase or decrease tendencies toward synchronous discharges between α-MNs. This controversy is not easy to settle experimentally in animal or human paradigms because RCs receive, in addition to excitatory input from α-MNs, many other modulating influences which may change their mode of operation. Computer simulations help to artificially isolate the recurrent inhibitory circuit and thus to study its effects on α-MN synchronization under conditions not achievable in natural experiments. We present here such a study which was designed to specifically test the following hypothesis. Since many α-MNs excite any particular Renshaw cell, which in turn inhibits many α-MNs, this convergence–divergence pattern establishes a random network whose random discharge patterns inject uncorrelated noise into α-MNs, and this noise counteracts any synchronization potentially arising from other sources, e.g., common inputs (Adam et al. in Biol Cybern 29:229–235, 1978). We investigated the short-term synchronization of α-MNs with two types of excitatory input signals to α-MNs (random and sinusoidally modulated random patterns). The main results showed that, while recurrent inhibitory inputs to different α-MNs were indeed different, recurrent inhibition (1) exerted rather small effects on the modulation of α-MN discharge, (2) tended to increase the short-term synchronization of α-MN discharge, and (3) did not generate secondary peaks in α-MN-α-MN cross-correlograms associated with α-MN rhythmicity.     

Inhibitory Control of Linear and Supralinear Dendritic Excitation in CA1 Pyramidal Neurons

The transformation of dendritic excitatory synaptic inputs to axonal action potential output is the fundamental computation performed by all principal neurons. We show that in the hippocampus this transformation is potently controlled by recurrent inhibitory microcircuits. However, excitatory input on highly excitable dendritic branches could resist inhibitory?control by generating strong dendritic spikes and?trigger precisely timed action potential output. Furthermore, we show that inhibition-sensitive branches can be transformed into inhibition-resistant, strongly spiking branches by intrinsic plasticity of branch excitability. In addition, we demonstrate that the inhibitory control of spatially defined dendritic excitation is strongly regulated by network activity patterns. Our findings suggest that dendritic spikes may serve to transform correlated branch input into reliable and temporally precise output even in the presence of inhibition.     

Distributed dendritic processing facilitates object detection: a computational analysis on the visual system of the fly

Background

Detecting objects is an important task when moving through a natural environment. Flies, for example, may land on salient objects or may avoid collisions with them. The neuronal ensemble of Figure Detection cells (FD-cells) in the visual system of the fly is likely to be involved in controlling these behaviours, as these cells are more sensitive to objects than to extended background structures. Until now the computations in the presynaptic neuronal network of FD-cells and, in particular, the functional significance of the experimentally established distributed dendritic processing of excitatory and inhibitory inputs is not understood.

Methodology/Principal Findings

We use model simulations to analyse the neuronal computations responsible for the preference of FD-cells for small objects. We employed a new modelling approach which allowed us to account for the spatial spread of electrical signals in the dendrites while avoiding detailed compartmental modelling. The models are based on available physiological and anatomical data. Three models were tested each implementing an inhibitory neural circuit, but differing by the spatial arrangement of the inhibitory interaction. Parameter optimisation with an evolutionary algorithm revealed that only distributed dendritic processing satisfies the constraints arising from electrophysiological experiments. In contrast to a direct dendro-dendritic inhibition of the FD-cell (Direct Distributed Inhibition model), an inhibition of its presynaptic retinotopic elements (Indirect Distributed Inhibition model) requires smaller changes in input resistance in the inhibited neurons during visual stimulation.

Conclusions/Significance

Distributed dendritic inhibition of retinotopic elements as implemented in our Indirect Distributed Inhibition model is the most plausible wiring scheme for the neuronal circuit of FD-cells. This microcircuit is computationally similar to lateral inhibition between the retinotopic elements. Hence, distributed inhibition might be an alternative explanation of perceptual phenomena currently explained by lateral inhibition networks.     

Response of Renshaw cells to sinusoidal stretch of hindlimb extensor muscles.

1. Renshaw cells responding disynaptically to electrically induced group I volleys in the intact gastrocnemius-soleus (GS) nerve, were submitted to small-amplitude, high-frequency vibration applied longitudinally to the deefferented GS muscle in precollicular decerebrate cats. 2. Vibration of the GS muscle at 200/sec, 180 mu peak-to-peak amplitude for 80-100 msec produced a sudden increase in the discharge rate of Renshaw cells, which gradually decreased within 25-50 msec to reach a steady level higher than that recorded in the absence of vibration. 3. Excitation of Renshaw cells appeared at a threshold amplitude of vibration (at 200-250/sec) of 5-20 mu and increased to a maximum value for amplitudes of about 70-80 mu, i.e., when all the primary endings of the spindles from the GS muscle had been driven by the stimulus. Recruitment of the secondary endings of the muscle spindles, due to large amplitude muscle vibration, did not modify the response of the Renshaw cells to the mechanically induced group Ia volleys. 4. These findings were obtained with the GS muscle pulled at 8 mm of initial extension. A threshold response of Renshaw cells to vibration appeared at 4 mm of static stretch, while maximal responses occurred at 8 mm. No further increase and actually a slight decrease in the response appeared for initial extensions of the muscle of 10-12 mm. 5. For a given vibration amplitude, the response of the Renshaw cells increased with increasing frequencies of vibration to reach the maximum at frequencies of 150-250/sec. Bursts of Renshaw cell discharges synchronous to each stroke of vibrator occurred only for low frequencies of stimulation (less than 25/sec). 6. It is concluded that vibration of the GS muscle represents a very effective method in exciting the Renshaw cells and that this response depends upon selective stimulation of homonymous motoneurons monosynaptically excited by the orthodromic volleys originating from the primary endings of the corresponding muscle spindles.     

Model studies of lateral inhibition as a mechanism of detecting motion. II. Modeling feedback inhibition

A program is worked out which imitates non-stationary lateral inhibition in two-dimensional retina with physiologically real characteristics. Experiments with moving and flashing stimuli of different size are carried out. It is shown that lateral inhibition may serve as a mechanism which forms selective sensitivity of the elements involved to the movement of the stimuli. Recurrent inhibition proves to be more effective than the direct one providing that a longer delay in time exists. The results obtained are explained by the peculiarities of recurrent inhibition as the feedback system.     

System principles in the organization of neurons carrying out cortical inhibition

Some new data on neuronal and synaptic organization of sensorimotor cortical area in cat are obtained by a complex of morphological and electrophysiological methods. These data permit considering that direct afferent inhibition is ensured by thalamo-cortical neurons and neurons forming the callosal and association links. The recurrent and lateral inhibition are structurally realized through the ascending recurrent axon collaterals of pyramidal neurons forming links either with short-axon or with long-axon interneurons. Cortico-thalamic (cortico-fugal) inhibition may be performed either via descending cortico-thalamic neurons or via cortico-cortical ipsi- and contralateral neurons. The above mentioned neuronal chains may be considered as structural elements of more complex neuronal sets which ensure the inhibition at the cortical inputs, outputs and intracortically.     

The demonstration of recurrent motor axon collaterals in the chick embryo by using horseradish peroxidase axonal transport]

Motoneurons were labelled by retrograde axonal transport of HRP applied to transected spinal nerves in 9-11-day chick embryos in the in vitro spinal cord preparation. Recurrent motor axon collaterals were revealed in 17 of 48 motor axons which could be followed in the edge regions of labelled motoneuronal pools. The results, coupled with author's earlier electrophysiological data, provide further evidence for the presence of the Renshaw inhibition in the avian spinal cord.     

Simulation system of spinal cord motor nuclei and associated nerves and muscles,in a Web-based architecture

A Web-based simulation system of the spinal cord circuitry responsible for muscle control is described. The simulator employs two-compartment motoneuron models for S, FR and FF types, with synaptic inputs acting through conductance variations. Four motoneuron pools with their associated interneurons are represented in the simulator, with the possibility of inclusion of more than 2,000 neurons and 2,000,000 synapses. Each motoneuron action potential is followed, after a conduction delay, by a motor unit potential and a motor unit twitch. The sums of all motor unit potentials and twitches result in the electromyogram (EMG), and the muscle force, respectively. Inputs to the motoneuron pool come from populations of interneurons (Ia reciprocal inhibitory interneurons, Ib interneurons, and Renshaw cells) and from stochastic point processes associated with descending tracts. To simulate human electrophysiological experiments, the simulator incorporates external nerve stimulation with orthodromic and antidromic propagation. This provides the mechanisms for reflex generation and activation of spinal neuronal circuits that modulate the activity of another motoneuron pool (e.g., by reciprocal inhibition). The generation of the H-reflex by the Ia-motoneuron pool system and its modulation by spinal cord interneurons is included in the simulation system. Studies with the simulator may include the statistics of individual motoneuron or interneuron spike trains or the collective effect of a motor nucleus on the dynamics of muscle force control. Properties associated with motor-unit recruitment, motor-unit synchronization, recurrent inhibition and reciprocal inhibition may be investigated.