How shunting inhibition affects the discharge of lumbar motoneurones: a dynamic clamp study in anaesthetized cats

A strategy for mapping biophysical to abstract neuronal network models applied to primary visual cortex

A fundamental challenge for the theoretical study of neuronal networks is to make the link between complex biophysical models based directly on experimental data, to progressively simpler mathematical models that allow the derivation of general operating principles. We present a strategy that successively maps a relatively detailed biophysical population model, comprising conductance-based Hodgkin-Huxley type neuron models with connectivity rules derived from anatomical data, to various representations with fewer parameters, finishing with a firing rate network model that permits analysis. We apply this methodology to primary visual cortex of higher mammals, focusing on the functional property of stimulus orientation selectivity of receptive fields of individual neurons. The mapping produces compact expressions for the parameters of the abstract model that clearly identify the impact of specific electrophysiological and anatomical parameters on the analytical results, in particular as manifested by specific functional signatures of visual cortex, including input-output sharpening, conductance invariance, virtual rotation and the tilt after effect. Importantly, qualitative differences between model behaviours point out consequences of various simplifications. The strategy may be applied to other neuronal systems with appropriate modifications.

A hierarchy of theoretical approaches to study a neuronal network depends on a tradeoff between biological fidelity and mathematical tractibility. Biophysically-detailed models consider cellular mechanisms and anatomically defined synaptic circuits, but are often too complex to reveal insights into fundamental principles. In contrast, increasingly abstract reduced models facilitate analytical insights. To better ground the latter to the underlying biology, we describe a systematic procedure to move across the model hierarchy that allows understanding how changes in biological parameters—physiological, pathophysiological, or because of new data—impact the behaviour of the network. We apply this approach to mammalian primary visual cortex, and examine how the different models in the hierarchy reproduce functional signatures of this area, in particular the tuning of neurons to the orientation of a visual stimulus. Our work provides a navigation of the complex parameter space of neural network models faithful to biology, as well as highlighting how simplifications made for mathematical convenience can fundamentally change their behaviour.

Emergent Elements of Inspiratory Rhythmogenesis: Network Synchronization and Synchrony Propagation

We assessed the mechanism of mammalian breathing rhythmogenesis in the preBötzinger complex (preBötC) in vitro, where experimental tests remain inconsistent with hypotheses of canonical rhythmogenic cellular or synaptic mechanisms, i.e., pacemaker neurons or inhibition. Under rhythmic conditions, in each cycle, an inspiratory burst emerges as (presumptive) preBötC rhythmogenic neurons transition from aperiodic uncorrelated population spike activity to become increasingly synchronized during preinspiration (for ∼50-500 ms), which can trigger inspiratory bursts that propagate to motoneurons. In nonrhythmic conditions, antagonizing GABA_A receptors can initiate this synchronization while inducing a higher conductance state in nonrhythmogenic preBötC output neurons. Our analyses uncover salient features of preBötC network dynamics where inspiratory bursts arise when and only when the preBötC rhythmogenic subpopulation strongly synchronizes to drive output neurons. Furthermore, downstream propagation of preBötC network activity, ultimately to motoneurons, is dependent on the strength of input synchrony onto preBötC output neurons exemplifying synchronous propagation of network activity.

The influence of increased membrane conductance on response properties of spinal motoneurons

During functional spinal neural network activity motoneurons receive massive synaptic excitation and inhibition, and their membrane conductance increases considerably - they are switched to a high-conductance state. High-conductance states can substantially alter response properties of motoneurons. In the present study we investigated how an increase in membrane conductance affects spike frequency adaptation, the gain (i.e., the slope of the frequency-current relationship) and the threshold for action potential generation. We used intracellular recordings from adult turtle motoneurons in spinal cord slices. Membrane conductance was increased pharmacologically by extracellular application of the GABAA receptor agonist muscimol. Our findings suggest that an increase in membrane conductance of about 40-50% increases the magnitude of spike frequency adaptation, but does not change the threshold for action potential generation. Increased conductance causes a subtractive rather than a divisive effect on the initial and the early frequency-current relationships and may have not only a subtractive but also a divisive effect on the steady-state frequency-current relationship.

The domain of neuronal firing on a plane of input current and conductance

The activation of neurotransmitter receptors increases the current flow and membrane conductance and thus controls the firing rate of a neuron. In the present work, we justified the two-dimensional representation of a neuronal input by voltage-independent current and conductance and obtained experimentally and numerically a complete input-output (I/O) function. The dependence of the steady-state firing rate on the input current and conductance was studied as a two-parameter I/O function. We employed the dynamic patch clamp technique in slices to get this dependence for the whole domain of two input signals that evoke stationary spike trains in a single neuron (Ω-domain). As found, the Ω-domain is finite and an additional conductance decreases the range of spike-evoking currents. The I/O function has been reproduced in a Hodgkin-Huxley-like model. Among the simulated effects of different factors on the I/O function, including passive and active membrane properties, external conditions and input signal properties, the most interesting were: the shift of the right boundary of the Ω-domain (corresponding to the exCitation block) leftwards due to the decrease of the maximal potassium conductance; and the reduction of the Ω-domain by the decrease of the maximal sodium concentration. As found in experiments and simulations, the Ω-domain is reduced by the decrease of extracellular sodium concentration, by cooling, and by adding slow potassium currents providing interspike interval adaptation; the Ω-domain height is increased by adding color noise. Our modeling data provided a generalization of I/O dependencies that is consistent with previous studies and our experiments. Our results suggest that both current flow and membrane conductance should be taken into account when determining neuronal firing activity.

Mechanisms of gain control by voltage-gated channels in intrinsically-firing neurons

Gain modulation is a key feature of neural information processing, but underlying mechanisms remain unclear. In single neurons, gain can be measured as the slope of the current-frequency (input-output) relationship over any given range of inputs. While much work has focused on the control of basal firing rates and spike rate adaptation, gain control has been relatively unstudied. Of the limited studies on gain control, some have examined the roles of synaptic noise and passive somatic currents, but the roles of voltage-gated channels present ubiquitously in neurons have been less explored. Here, we systematically examined the relationship between gain and voltage-gated ion channels in a conductance-based, tonically-active, model neuron. Changes in expression (conductance density) of voltage-gated channels increased (Ca²⁺ channel), reduced (K⁺ channels), or produced little effect (h-type channel) on gain. We found that the gain-controlling ability of channels increased exponentially with the steepness of their activation within the dynamic voltage window (voltage range associated with firing). For depolarization-activated channels, this produced a greater channel current per action potential at higher firing rates. This allowed these channels to modulate gain by contributing to firing preferentially at states of higher excitation. A finer analysis of the current-voltage relationship during tonic firing identified narrow voltage windows at which the gain-modulating channels exerted their effects. As a proof of concept, we show that h-type channels can be tuned to modulate gain by changing the steepness of their activation within the dynamic voltage window. These results show how the impact of an ion channel on gain can be predicted from the relationship between channel kinetics and the membrane potential during firing. This is potentially relevant to understanding input-output scaling in a wide class of neurons found throughout the brain and other nervous systems.

Modulation of the input-output function by GABAA receptor-mediated currents in rat oculomotor nucleus motoneurons

The neuronal input-output function depends on recruitment threshold and gain of the firing frequency-current (f-I) relationship. These two parameters are positively correlated in ocular motoneurons (MNs) recorded in alert preparation and inhibitory inputs could contribute to this correlation. Phasic inhibition mediated by γ-amino butyric acid (GABA) occurs when a high concentration of GABA at the synaptic cleft activates postsynaptic GABAA receptors, allowing neuronal information transfer. In some neuronal populations, low concentrations of GABA activate non-synaptic GABAA receptors and generate a tonic inhibition, which modulates cell excitability. This study determined how ambient GABA concentrations modulate the input-output relationship of rat oculomotor nucleus MNs. Superfusion of brain slices with GABA (100 μm) produced a GABAA receptor-mediated current that reduced the input resistance, increased the recruitment threshold and shifted the f-I relationship rightward without any change in gain. These modifications did not depend on MN size. In absence of exogenous GABA, gabazine (20 μm; antagonist of GABAA receptors) abolished spontaneous inhibitory postsynaptic currents and revealed a tonic current in MNs. Gabazine increased input resistance and decreased recruitment threshold mainly in larger MNs. The f-I relationship shifted to the left, without any change in gain. Gabazine effects were chiefly due to MN tonic inhibition because tonic current amplitude was five-fold greater than phasic. This study demonstrates a tonic inhibition in ocular MNs that modulates cell excitability depending on cell size. We suggest that GABAA tonic inhibition acting concurrently with glutamate receptors activation could reproduce the positive covariation between threshold and gain reported in alert preparation.

The dendritic location of the L-type current and its deactivation by the somatic AHP current both contribute to firing bistability in motoneurons

Spinal motoneurons may display a variety of firing patterns including bistability between repetitive firing and quiescence and, more rarely, bistability between two firing states of different frequencies. It was suggested in the past that firing bistability required that the persistent L-type calcium current be segregated in distal dendrites, far away from the spike generating currents. However, this is not supported by more recent data. Using a two compartment model of motoneuron, we show that the different firing patterns may also result from the competition between the more proximal dendritic component of the dendritic L-type conductance and the calcium sensitive potassium conductance responsible for afterhypolarization (AHP). Further emphasizing this point, firing bistability may be also achieved when the L-type current is put in the somatic compartment. However, this requires that the calcium-sensitive potassium conductance be triggered solely by the high threshold calcium currents activated during spikes and not by calcium influx through the L-type current. This prediction was validated by dynamic clamp experiments in vivo in lumbar motoneurons of deeply anesthetized cats in which an artificial L-type current was added at the soma. Altogether, our results suggest that the dynamical interaction between the L-type and afterhyperpolarization currents is as fundamental as the segregation of the calcium L-type current in dendrites for controlling the discharge of motoneurons.

Synaptic and extrasynaptic plasticity in glutamatergic circuits involving dentate granule cells following chronic N-methyl-D-aspartate receptor inhibition

Chronic global N-methyl-d-aspartate receptor (NMDAR) blockade leads to changes in glutamatergic transmission. The impact of more subunit-selective NMDAR inhibition on glutamatergic circuits remains incomplete. To this end, organotypic hippocampal slice cultures were treated for 17-21 days with the high-affinity competitive antagonist d-aminophosphonovaleric acid (d-APV), the allosteric GluN2B-selective antagonist Ro25-6981, or the newer competitive GluN2A-preferring antagonist NVP-AAM077. Electrophysiological recordings from dentate granule cells revealed that chronic d-APV treatment increased, whereas chronic Ro25-6981 reduced, epileptiform event-associated large-amplitude spontaneous excitatory postsynaptic currents (sEPSC) compared with all other treatment groups, consistent with opposite effects on glutamatergic networks. Presynaptically, chronic d-APV or Ro25-6981 increased small-amplitude sEPSCs and AMPA/kainate receptor-mediated miniature EPSCs (mEPSCAMPAR) frequency. Chronic d-APV or NVP-AAM077, but not Ro25-6981, increased putative vGlut1-positive glutamatergic synapses. Postsynaptically, chronic d-APV dramatically increased mEPSCAMPAR and profoundly decreased NMDAR-mediated mEPSC (mEPSCNMDAR) measures, suggesting increased AMPAR/NMDAR ratio. Ro25-6981 decreased mEPSCAMPAR charge transfer and modestly decreased mEPSCNMDAR frequency and decay, suggesting downward scaling of AMPAR and NMDAR function without dramatically altering AMPAR/NMDAR ratio. Extrasynaptically, threo-β-benzyloxyaspartate-enhanced "tonic" NMDAR current amplitude and activated channel number estimates were significantly increased only by chronic Ro25-6981. For intrinsic excitability, action potential threshold was slightly more negative following chronic d-APV or NVP-AAM077. The predominant pro-excitatory effects of chronic d-APV are consistent with increased glutamatergic transmission and network excitability. The minor effects of chronic NVP-AAM077 on action potential threshold and synapse number are consistent with minimal effects on circuit function. The chronic Ro25-6981-induced downward scaling of synaptic AMPAR and NMDAR function is consistent with decreased postsynaptic glutamate receptors and reduced network excitability.

Statistical computer model analysis of the reciprocal and recurrent inhibitions of the Ia-EPSP in α-motoneurons

We simulate the inhibition of Ia-glutamatergic excitatory postsynaptic potential (EPSP) by preceding it with glycinergic recurrent (REN) and reciprocal (REC) inhibitory postsynaptic potentials (IPSPs). The inhibition is evaluated in the presence of voltage-dependent conductances of sodium, delayed rectifier potassium, and slow potassium in five α-motoneurons (MNs). We distribute the channels along the neuronal dendrites using, alternatively, a density function of exponential rise (ER), exponential decay (ED), or a step function (ST). We examine the change in EPSP amplitude, the rate of rise (RR), and the time integral (TI) due to inhibition. The results yield six major conclusions. First, the EPSP peak and the kinetics depending on the time interval are either amplified or depressed by the REC and REN shunting inhibitions. Second, the mean EPSP peak, its TI, and RR inhibition of ST, ER, and ED distributions turn out to be similar for analogous ranges of G. Third, for identical G, the large variations in the parameters' values can be attributed to the sodium conductance step (g(Na_step)) and the active dendritic area. We find that small g(Na_step) on a few dendrites maintains the EPSP peak, its TI, and RR inhibition similar to the passive state, but high g(Na_step) on many dendrites decrease the inhibition and sometimes generates even an excitatory effect. Fourth, the MN's input resistance does not alter the efficacy of EPSP inhibition. Fifth, the REC and REN inhibitions slightly change the EPSP peak and its RR. However, EPSP TI is depressed by the REN inhibition more than the REC inhibition. Finally, only an inhibitory effect shows up during the EPSP TI inhibition, while there are both inhibitory and excitatory impacts on the EPSP peak and its RR.

Dopaminergic Control of the Exploration-Exploitation Trade-Off via the Basal Ganglia

We continuously face the dilemma of choosing between actions that gather new information or actions that exploit existing knowledge. This "exploration-exploitation" trade-off depends on the environment: stability favors exploiting knowledge to maximize gains; volatility favors exploring new options and discovering new outcomes. Here we set out to reconcile recent evidence for dopamine's involvement in the exploration-exploitation trade-off with the existing evidence for basal ganglia control of action selection, by testing the hypothesis that tonic dopamine in the striatum, the basal ganglia's input nucleus, sets the current exploration-exploitation trade-off. We first advance the idea of interpreting the basal ganglia output as a probability distribution function for action selection. Using computational models of the full basal ganglia circuit, we showed that, under this interpretation, the actions of dopamine within the striatum change the basal ganglia's output to favor the level of exploration or exploitation encoded in the probability distribution. We also found that our models predict striatal dopamine controls the exploration-exploitation trade-off if we instead read-out the probability distribution from the target nuclei of the basal ganglia, where their inhibitory input shapes the cortical input to these nuclei. Finally, by integrating the basal ganglia within a reinforcement learning model, we showed how dopamine's effect on the exploration-exploitation trade-off could be measurable in a forced two-choice task. These simulations also showed how tonic dopamine can appear to affect learning while only directly altering the trade-off. Thus, our models support the hypothesis that changes in tonic dopamine within the striatum can alter the exploration-exploitation trade-off by modulating the output of the basal ganglia.

Mixed mode oscillations in mouse spinal motoneurons arise from a low excitability state

We explain the mechanism that elicits the mixed mode oscillations (MMOs) and the subprimary firing range that we recently discovered in mouse spinal motoneurons. In this firing regime, high-frequency subthreshold oscillations appear a few millivolts below the spike voltage threshold and precede the firing of a full blown spike. By combining intracellular recordings in vivo (including dynamic clamp experiments) in mouse spinal motoneurons and modeling, we show that the subthreshold oscillations are due to the spike currents and that MMOs appear each time the membrane is in a low excitability state. Slow kinetic processes largely contribute to this low excitability. The clockwise hysteresis in the I-F relationship, frequently observed in mouse motoneurons, is mainly due to a substantial slow inactivation of the sodium current. As a consequence, less sodium current is available for spiking. This explains why a large subprimary range with numerous oscillations is present in motoneurons displaying a clockwise hysteresis. In motoneurons whose I-F curve exhibits a counterclockwise hysteresis, it is likely that the slow inactivation operates on a shorter time scale and is substantially reduced by the de-inactivating effect of the afterhyperpolarization (AHP) current, thus resulting in a more excitable state. This accounts for the short subprimary firing range with only a few MMOs seen in these motoneurons. Our study reveals a new role for the AHP current that sets the membrane excitability level by counteracting the slow inactivation of the sodium current and allows or precludes the appearance of MMOs.

Gain control in CA1 pyramidal cells using changes in somatic conductance

Gain modulation is an important feature of neural activity. Previous work has focused on the ability of background synaptic noise to modulate the slope (i.e., gain) of the frequency-current (f-I) relationship in neurons. To date, demonstrations of gain control that are independent of synaptic noise have been limited. We investigated the effects of increasing somatic conductance in the form of tonic inhibition on the initial and steady-state f-I relationship of CA1 pyramidal cells. We find that increasing membrane conductance reduces the gain of the steady-state f-I relationship through a graded increase in the magnitude of spike frequency adaptation. Increased adaptation arises through a conductance-induced depolarization of spike voltage threshold. Thus, by increasing the magnitude of spike frequency adaptation, added conductance can reduce the gain of the steady-state f-I relationship in the absence of random background membrane fluctuations.

Exploring modulation of action potential firing by artificial graft of fast GABAergic autaptic afferences in hypophyseal neuroendocrine melanotrope cells

Excitable cells are connected via electrical and chemical synapses to form a complex plastic network that transmit and modify action potential trains. In vivo and in vitro studies suggest that powerful type A GABAergic self-innervations, known as autapses, play a central role in the shaping of spike discharges. Herein, we have investigated the effects of artificial autaptic activity on action potential firing in cultured hypophyseal neuroendocrine melanotrope cells removed from any synaptic input. Type A autaptic conductances were introduced by using a dedicated dynamic-clamp device, based on a digital signal processor. The results indicate that cells grafted with autaptic dynamic-clamp are subjected to a modulation of both evoked firing and artificial GABA(A)-currents. The autaptic feedback reduced the action potentials amplitude, decreasing both the overshoot and the after-hyperpolarization potential (AHP). In return, the reduced voltage changes diminished the autaptic current amplitudes. The overall effect was a decrease of the cell firing rate. The introduction of a variable autaptic delay strongly altered the cell responses. Under certain conditions, the artificial autaptic current formed an additional component of the AHP which was markedly augmented. This action resulted in a stabilization of the action potential train leading to a more regular firing. In conclusion, it appeared that the autaptic feedback was very dependent on functional parameters such as the autaptic distance. Further investigations are in progress to complete our model, taking the autaptic plasticity into account.

Fast kinetics, high-frequency oscillations, and subprimary firing range in adult mouse spinal motoneurons

The fast contraction time of mouse motor units creates a unique situation in which motoneurons have to fire at low frequencies to produce small forces but also at very high frequency (much higher than in cat or rat motoneurons) to reach the fusion frequency of their motor units. To understand how this problem is solved, we performed intracellular recordings of adult mouse spinal motoneurons and investigated systematically their subthreshold properties and their discharge pattern. We show that mouse motoneurons have a much wider range of firing frequencies than cat and rat motoneurons because of three salient features. First, they have a short membrane time constant. This results in a higher cutoff frequency and a higher resonance frequency, which allow mouse motoneurons to integrate inputs at higher frequencies. Second, their afterhyperpolarization (AHP) is faster, allowing the motoneurons to discharge at a higher rate. Third, motoneurons display high-frequency (100-150 Hz) subthreshold oscillations during the interspike intervals. The fast membrane kinetics greatly favors the appearance of these oscillations, creating a "subprimary range" of firing. In this range, which has never been reported in cat and in rat spinal motoneurons, the oscillations follow the AHP and trigger spiking after a variable delay, allowing a discharge at low frequency but at the expense of an irregular rate.

Causes of transient instabilities in the dynamic clamp

The dynamic clamp is a widely used method for integrating mathematical models with electrophysiological experiments. This method involves measuring the membrane voltage of a cell, using it to solve computational models of ion channel dynamics in real-time, and injecting the calculated current(s) back into the cell. Limitations of this technique include those associated with single electrode current clamping and the sampling effects caused by the dynamic clamp. In this study, we show that the combination of these limitations causes transient instabilities under certain conditions. Through physical experiments and simulations, we show that dynamic clamp instability is directly related to the sampling delay and the maximum simulated conductance being injected. It is exaggerated by insufficient electrode series resistance and capacitance compensation. Increasing the sampling rate of the dynamic clamp system increases dynamic clamp stability; however, this improvement, is constrained by how well the electrode series resistance and capacitance are compensated. At present, dynamic clamp sampling rates are justified solely on the temporal dynamics of the models being simulated; here we show that faster rates increase the stable range of operation for the dynamic clamp system. In addition, we show that commonly accepted levels of resistance compensation nevertheless significantly compromise the stability of a dynamic clamp system.

MAP estimation algorithm for phase response curves based on analysis of the observation process

Many research groups have sought to measure phase response curves (PRCs) from real neurons. However, methods of estimating PRCs from noisy spike-response data have yet to be established. In this paper, we propose a Bayesian approach for estimating PRCs. First, we analytically obtain a likelihood function of the PRC from a detailed model of the observation process formulated as Langevin equations. Then we construct a maximum a posteriori (MAP) estimation algorithm based on the analytically obtained likelihood function. The MAP estimation algorithm derived here is equivalent to the spherical spin model. Moreover, we analytically calculate a marginal likelihood corresponding to the free energy of the spherical spin model, which enables us to estimate the hyper-parameters, i.e., the intensity of the Langevin force and the smoothness of the prior.

Resonant or not, two amplification modes of proprioceptive inputs by persistent inward currents in spinal motoneurons

Why do motoneurons possess two persistent inward currents (PICs), a fast sodium current and a slow calcium current? To answer this question, we replaced the natural PICs with dynamic clamp-imposed artificial PICs at the soma of spinal motoneurons of anesthetized cats. We investigated how PICs with different kinetics (1-100 ms) amplify proprioceptive inputs. We showed that their action depends on the presence or absence of a resonance created by the I(h) current. In resonant motoneurons, a fast PIC enhances the resonance and amplifies the dynamic component of Ia inputs elicited by ramp-and-hold muscle stretches. This facilitates the recruitment of these motoneurons, which likely innervate fast contracting motor units developing large forces, e.g., to restore balance or produce ballistic movements. In nonresonant motoneurons, in contrast, a fast PIC easily triggers plateau potentials, which leads to a dramatic amplification of the static component of Ia inputs. This likely facilitates the recruitment of these motoneurons, innervating mostly slowly contracting and fatigue-resistant motor units, during postural activities. Finally, a slow PIC may switch a resonant motoneuron to nonresonant by counterbalancing I(h), thus changing the action of the fast PIC. A modeling study shows that I(h) needs to be located on the dendrites to create the resonance, and it predicts that dendritic PICs amplify synaptic input in the same manner as somatic PICs.

Dynamic clamp analyses of cardiac, endocrine, and neural function

The dynamic clamp introduces artificial conductances into cells to simulate electrical coupling, votage-dependent, leak, and synaptic conductances. This review describes how the dynamic clamp has been used to address various questions in the cardiac, endocrine, and nervous systems.

Direct control of firing rate gain by dendritic shunting inhibition

The firing rate gain of neurons, defined as the slope of the relation between input to a neuron and its firing rate, has received considerable attention in the past few years. This has been largely motivated by the many experimental demonstrations of behavior related gain changes in a variety of neural circuits of the CNS. A surprising result was that a prime candidate, shunting inhibition, apparently does not change the firing rate gain of neurons. However, in this paper, we show a physiologically plausible mechanism by which shunting inhibition in the dendritic tree does, in a simple and direct manner, modulate the firing gain of neurons. The effect is due to a strong attenuation of the dendritic current arriving at the soma by shunting dendritic inhibition. Increasing the dendritic inhibitory conductance enhances the attenuation of current flowing from the dendritic to the somatic compartment and thus reduces firing gain. This mechanism relies on known physiological and anatomical properties of CNS neurons and does not require special features such as tunable neural noise inputs. Gain control by the proposed mechanism may prove to be a ubiquitous feature of neural circuit operations and it is readily verifiable experimentally.

The afterhyperpolarization conductance exerts the same control over the gain and variability of motoneurone firing in anaesthetized cats

Does the afterhyperpolarization current control the gain and discharge variability of motoneurones according to the same law? We investigated this issue in lumbar motoneurones of anaesthetized cats. Using dynamic clamp, we measured the conductance, time constant and driving force of the AHP current in a sample of motoneurones and studied how the gain was correlated to these quantities. To study the action of the AHP on the discharge variability and to compare it to its action on the gain, we injected an artificial AHP-like current in motoneurones. This increased the natural AHP. In three motoneurones, we abolished most of the natural AHP with the calcium chelator BAPTA to investigate the condition where the discharge was essentially controlled by the artificial AHP. Our results demonstrate that both the gain and the coefficient of variation of the firing rate are inversely proportional to the magnitude and to the time constant of the artificial AHP conductance. This indicates that the AHP exerts the same control over the gain and the variability. This mechanism ensures that the variability of the discharge is modulated with the gain. This guarantees a great regularity of the discharge when the motoneurone is in a low excitability state and hence good control of the force produced.

Linear summation of cat motor cortex outputs

Recruitment of movement-related muscle synergies involves the functional linking of motor cortical points. We asked how the outputs of two simultaneously stimulated motor cortical points would interact. To this end, experiments were done in ketamine-anesthetized cats. When prolonged (e.g., 500 ms) trains of intracortical microstimulation were applied in the primary motor cortex, stimulus currents as low as 10-20 microA evoked coordinated movements of the contralateral forelimb. Paw kinematics in three dimensions and the electromyographic (EMG) activity of eight muscles were simultaneously recorded. We show that the EMG outputs of two cortical points simultaneously stimulated are additive. The movements were represented as displacement vectors pointing from initial to final paw position. The displacement vectors resulting from simultaneous stimulation of two cortical points pointed in nearly the same direction as the algebraic resultant vector. Linear summation of outputs was also found when inhibition at one of the cortical points was reduced by GABAA receptor antagonists. A simple principle emerges from these results. Notwithstanding the underlying complex neuronal circuitry, motor cortex outputs combine nearly linearly in terms of movement direction and muscle activation patterns. Importantly, simultaneous activation does not change the nature of the output at each point. An additional implication is that not all possible movements need be explicitly represented in the motor cortex; a large number of different movements may be synthesized from a smaller repertoire.

How much afterhyperpolarization conductance is recruited by an action potential? A dynamic-clamp study in cat lumbar motoneurons

We accurately measured the conductance responsible for the afterhyperpolarization (medium AHP) that follows a single spike in spinal motoneurons of anesthetized cats. This was done by using the dynamic-clamp method. We injected an artificial current in the neurons that increased the AHP amplitude, and we made use of the fact that the intensity of the natural AHP current at the trough of the voltage trajectory was related linearly to the AHP amplitude. We determined at the same time the conductance and the reversal potential of the AHP current. This new method was validated by a simple theoretical model incorporating AHP and hyperpolarization-activated (Ih) currents and could be applied when the decay time constant of the AHP conductance was at least five times shorter than the estimated Ih activation time. This condition was fulfilled in 33 of 44 motoneurons. The AHP conductance varied from 0.3 to 1.4 microS in both slow- and fast-type motoneurons, which was approximately the same range as the input conductance of the entire population. However, AHP and input conductances were not correlated. The larger AHP in slow-type motoneurons was mainly attributable to their smaller input conductance compared with fast motoneurons. The likeness of the AHP conductance in both types of motoneurons is in sharp contrast to differences in AHP decay time and explains why slow- and fast-type motoneurons have similar gain.

The Effect of NMDA Receptors on Gain Modulation

The ability of individual neurons to modulate the gain of their input-output function is important for information processing in the brain. In a recent study (Mitchell & Silver, 2003), shunting inhibition was found to modulate the gain of cerebellar granule cells subjected to simulated currents through AMPA receptor synapses. Here we investigate the effect on gain modulation resulting from adding the currents mediated by NMDA receptors to a compartmental model of the granule cell. With only AMPA receptors, the changes in gain induced by shunting inhibition decreased gradually with the average firing rate of the afferent mossy fibers. With NMDA receptors present, this decrease was more rapid, therefore narrowing the bandwidth of mossy fiber firing rates available for gain modulation. The deterioration of gain modulation was accompanied by a reduced variability of the input current and saturation of NMDA receptors. However, when the output of the granule cell was plotted as a function of the average input current instead of the input firing frequency, both models showed very similar response curves and comparable gain modulation. We conclude that NMDA receptors do not directly impair gain control by shunting inhibition, but the effective bandwidth decreases as a consequence of the increased total charge transfer.

How Membrane Properties Shape the Discharge of Motoneurons: A Detailed Analytical Study

Electrophysiological experiments and modeling studies have shown that after hyperpolarization regulates the discharge of lumbar motoneurons in anesthetized cats and is an important determinant of their firing properties. However, it is still unclear how firing properties depend on slow after hyperpolarization, input conductance, and the fast currents responsible for spike generation. We study a single-compartment integrate-andfire model with a slow potassium conductance that exponentially decays between spikes. We show that this model is analytically solvable, and we investigate how passive and active membrane properties control the discharge. We show that the model exhibits three distinct firing ranges (primary, secondary, and high frequency), and we explain the origin of these three ranges. Explicit expressions are established for the gain of the steady-state current-frequency (I− f) curve in the primary range and for the gain of the I− f curve for the first interspike interval. They show how the gain is controlled by the maximal conductance and the kinetic parameters of the after hyperpolarization conductance. The gain also depends on the spike voltage threshold, and we compute how it is decreased by threshold accommodation (i.e., the increase of the threshold with the injected current). In contrast, the gain is not very sensitive to the input conductance. This implies that tonic synaptic activity shifts the current-frequency curve in its primary range, in agreement with experiments. Taking into account the absolute refractory period associated with spikes somewhat reduces the gain in the primary range. More importantly, it accounts for the approximately linear and steep secondary range observed in many motoneurons. In the nonphysiological high-frequency range, the behavior of the I− f curve is determined by the fast voltage-dependent currents, via the amplitude of the fast repolarization afterspike, the duration of the refractory period, and voltage threshold accommodation, if present.

Comparison of the inhibition of Renshaw cells during subthreshold and suprathreshold conditions using anatomically and physiologically realistic models

Inhibitory synaptic inputs to Renshaw cells are concentrated on the soma and the juxtasomatic dendrites. In the present study, we investigated whether this proximal bias leads to more effective inhibition under different neuronal operating conditions. Using compartmental models based on detailed anatomical measurements of intracellularly stained Renshaw cells, we compared the inhibition produced by glycine/gamma-aminobutyric acid-A (GABA(A)) synapses when distributed with a proximal bias to the inhibition produced when the same synapses were distributed uniformly (i.e., with no regional bias). The comparison was conducted in subthreshold and suprathreshold conditions. The latter were mimicked by voltage clamping the soma to -55 mV. The voltage clamp reduces nonlinear interactions between excitatory and inhibitory synapses. We hypothesized that for electrotonically compact cells such as Renshaw cells, the strength of the inhibition would become much less dependent on synaptic location in suprathreshold conditions. This hypothesis was not confirmed. The inhibition produced when inhibitory inputs were proximally distributed was always stronger than when the same inputs were uniformly distributed. In fact, the relative effectiveness of proximally distributed inhibitory inputs over uniformly distributed synapses was greater in suprathreshold conditions than that in subthreshold conditions. The somatic voltage clamp minimized saturation of inhibitory driving potentials. Because this effect was greatest near the soma, the current produced by more distal synapses suffered a greater loss because of saturation. Conversely, in subthreshold conditions, the effectiveness of proximal synapses was substantially reduced at high levels of background synaptic activity because of saturation. Our results suggest glycine/GABA(A) synapses on Renshaw cells are strategically distributed to block the powerful excitatory drive produced by recurrent collaterals from motoneurons.

The combined effects of inhibitory and electrical synapses in synchrony

Recent experimental results have shown that GABAergic interneurons in the central nervous system are frequently connected via electrical synapses. Hence, depending on the area or the subpopulation, interneurons interact via inhibitory synapses or electrical synapses alone or via both types of interactions. The theoretical work presented here addresses the significance of these different modes of interactions for the interneuron networks dynamics. We consider the simplest system in which this issue can be investigated in models or in experiments: a pair of neurons, interacting via electrical synapses, inhibitory synapses, or both, and activated by the injection of a noisy external current. Assuming that the couplings and the noise are weak, we derive an analytical expression relating the cross-correlation (CC) of the activity of the two neurons to the phase response function of the neurons. When electrical and inhibitory interactions are not too strong, they combine their effect in a linear manner. In this regime, the effect of electrical and inhibitory interactions when combined can be deduced knowing the effects of each of the interactions separately. As a consequence, depending on intrinsic neuronal properties, electrical and inhibitory synapses may cooperate, both promoting synchrony, or may compete, with one promoting synchrony while the other impedes it. In contrast, for sufficiently strong couplings, the two types of synapses combine in a nonlinear fashion. Remarkably, we find that in this regime, combining electrical synapses with inhibition amplifies synchrony, whereas electrical synapses alone would desynchronize the activity of the neurons. We apply our theory to predict how the shape of the CC of two neurons changes as a function of ionic channel conductances, focusing on the effect of persistent sodium conductance, of the firing rate of the neurons and the nature and the strength of their interactions. These predictions may be tested using dynamic clamp techniques.