Two opposite voltage-dependent currents control the unusual early development pattern of embryonic Renshaw cell electrical activity.

Renshaw cells (V1R) are excitable as soon as they reach their final location next to the spinal motoneurons and are functionally heterogeneous. Using multiple experimental approaches, in combination with biophysical modeling and dynamical systems theory, we analyzed, for the first time, the mechanisms underlying the electrophysiological properties of V1R during early embryonic development of the mouse spinal cord locomotor networks (E11.5-E16.5). We found that these interneurons are subdivided into several functional clusters from E11.5 and then display an unexpected transitory involution process during which they lose their ability to sustain tonic firing. We demonstrated that the essential factor controlling the diversity of the discharge pattern of embryonic V1R is the ratio of a persistent sodium conductance to a delayed rectifier potassium conductance. Taken together, our results reveal how a simple mechanism, based on the synergy of two voltage-dependent conductances that are ubiquitous in neurons, can produce functional diversity in embryonic V1R and control their early developmental trajectory.

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.

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.

Extending cable theory to heterogeneous dendrites.

Dendrites may exhibit many types of electrical and morphological heterogeneities at the scale of a few micrometers. Models of neurons, even so-called detailed models, rarely consider such heterogeneities. Small-scale fluctuations in the membrane conductances and the diameter of dendrites are generally disregarded and spines merely incorporated into the dendritic shaft. Using the two-scales method known as homogenization, we establish explicit expressions for the small-scale fluctuations of the membrane voltage, and we derive the cable equation satisfied by the voltage when these fluctuations are averaged out. This allows us to rigorously establish under what conditions a heterogeneous dendrite can be approximated by a homogeneous cable. We consider different distributions of synapses, orderly or random, on a passive dendrite, and we investigate when replacing excitatory and inhibitory synaptic conductances by their local averages leads to a small error in the voltage. This indicates in which regimes the approximations made in compartmental models are justified. We extend these results to active membranes endowed with voltage-dependent conductances or NMDA receptors. Then we examine under which conditions a spiny dendrite behaves as a smooth dendrite. We discover a new regime where this holds true, namely, when the conductance of the spine neck is small compared to the conductance of the synapses impinging on the spine head. Spines can then be taken into account by an effective excitatory current, the capacitance of the dendrite remaining unchanged. In this regime, the synaptic current transmitted from a spine to the dendritic shaft is strongly attenuated by the weak coupling conductance, but the total current they deliver can be quite substantial. These results suggest that pedunculated spines and stubby spines might play complementary roles in synaptic integration. Finally, we analyze how varicosities affect voltage diffusion in dendrites and discuss their impact on the spatiotemporal integration of synaptic input.

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.

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.

How membrane properties shape the discharge of motoneurons: a detailed analytical study.

Electrophysiological experiments and modeling studies have shown that afterhyperpolarization 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 afterhyperpolarization, input conductance, and the fast currents responsible for spike generation. We study a single-compartment integrate-and-fire 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 afterhyperpolarization 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.

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.

Playing the devil's advocate: is the Hodgkin-Huxley model useful?

Hodgkin and Huxley (H-H) model for action potential generation has held firm for half a century because this relatively simple and experimentally testable model embodies the major features of membrane nonlinearity: namely, voltage-dependent ionic currents that activate and inactivate in time. However, experimental and theoretical developments of the past 20 years force one to re-evaluate its usefulness. First, the H-H model is, in its original form, limited to the two voltage-dependent currents found in the squid giant axon and it must be extended significantly if it is to deal with the excitable soma and dendrites of neurons. Second, the macroscopic and deterministic H-H model does not capture correctly the kinetics of the Na(+) channel and it cannot account for the stochastic response to current injection that arises from the discrete nature of ion channels. Third, much simpler integrate-and-fire-type models seem to be more useful for exploring collective phenomena in neuronal networks. Is the H-H model threatened, or will it continue to set the fundamental framework for exploring neuronal excitability?