This chapter summarizes a number of factors that control the "input-output" function across the motoneurons (MNs) comprising a single spinal motor nucleus. The main focus is on intrinsic properties of individual MNs that can be controlled by neuromodulators. These include: (1) amplification of the synaptic input at the cell's dendritic level by voltage-gated, persistent inward currents (plateau potentials); and (2) transduction of the net synaptic excitation into a frequency code (the MN's stimulus current-spike frequency relation) at the cell's soma/initial segment. Two other aspects of the synaptic control of MNs, which may affect their input-output gain, are also discussed. They include the hypotheses that: (1) a non-uniform distribution of synaptic effects to low- and high-threshold motor units causes a change in recruitment gain; and (2) recurrent inhibition, via motor axon collaterals and Renshaw cells, functions as a variable gain regulator of MN discharge.

A detailed knowledge of the quantitative properties of the currents I(Na) and I(K) underlying the action potential is essential for a deeper understanding of neuronal excitatory processes. However, it is not always possible or practical to perform voltage-clamp measurements that usually provide the necessary data. In this paper, we present a method by which the activation and kinetic properties of these currents can be estimated from current-clamp data, more precisely from the time course of the action potential, provided some additional electrophysiological properties of the neurone are a priori known. We report results from thalamocortical neurones and a cortical pyramidal cell, and suggest that the method will work with other types of neurones, if their action potentials are primarily shaped by I(Na) and I(K).

Uniform and non-uniform somato-dendritic distributions of the ion channels carrying the low-threshold Ca(2+) current (I(T)), the hyperpolarization-activated inward current (I(h)), the fast Na(+) current (I(Na)) and the delayed rectifier current (I(K)) were investigated in a multi-compartment model of a thalamocortical neuron for their suitability to reproduce the delta oscillation and the retinal excitatory post-synaptic potential recorded in vitro from the soma of thalamocortical neurons. The backpropagation of these simulated activities along the dendritic tree was also studied. A uniform somato-dendritic distribution of the maximal conductance of I(T) and I(K) (g(T) and g(K), respectively) was sufficient to simulate with acceptable accuracy: (i) the delta oscillation, and its phase resetting by somatically injected current pulses; as well as (ii) the retinal excitatory postsynaptic potential, and its alpha-amino-3-hydroxy-5-methyl-4-isoxazole proprionate and/or N-methyl-D-aspartate components. In addition, simulations where the dendritic g(T) and g(K) were either reduced (both by up to 34%) or increased (both by up to 15%) of their respective value on the soma still admitted a successful reproduction of the experimental activity. When the dendritic distributions were non-uniform, models where the proximal and distal dendritic g(T) was up to 1.8- and 1. 2-fold larger, respectively, than g(T(s)) produced accurate simulations of the delta oscillation (and its phase resetting curves) as well as the synaptic potentials without need of a concomitant increase in proximal or distal dendritic g(K). Furthermore, an increase in proximal dendritic g(T) and g(K) of up to fourfold their respective value on the soma resulted in acceptable simulation results. Addition of dendritic Na(+) channels to the uniformly or non-uniformly distributed somato-dendritic T-type Ca(2+) and K(+) channels did not further improve the overall qualitative and quantitative accuracy of the simulations, except for increasing the number of action potentials in bursts elicited by low-threshold Ca(2+) potentials. Dendritic I(h) failed to produce a marked effect on the simulated delta oscillation and the excitatory postsynaptic potential. In the presence of uniform and non-uniform dendritic g(T) and g(K), the delta oscillation propagated from the soma to the distal dendrites with no change in frequency and voltage-dependence, though the dendritic action potential amplitude was gradually reduced towards the distal dendrites. The amplitude and rising time of the simulated retinal excitatory postsynaptic potential were only slightly decreased during their propagation from their proximal dendritic site of origin to the soma or the distal dendrites. These results indicate that a multi-compartment model with passive dendrites cannot fully reproduce the experimental activity of thalamocortical neurons, while both uniform and non-uniform somato-dendritic g(T) and g(K) distributions are compatible with the properties of the delta oscillation and the retinal excitatory postsynaptic potential recorded in vitro from the soma of these neurons. Furthermore, by predicting the existence of backpropagation of low-threshold Ca(2+) potentials and retinal postsynaptic potentials up to the distal dendrites, our findings suggest a putative role for the delta oscillation in the dendritic processing of neuronal activity, and support previous hypotheses on the interaction between retinal and cortical excitatory postsynaptic potentials on thalamocortical neuron dendrites.

The propagation of excitation along the dendrites and the axon of a neurone is described by a partial differential equation which is nonlinear when voltage-gated conductances are present. In this case, numerical methods are employed to obtain a solution: the evolution of the membrane potential in space and time. Even when the membrane is passive (linear), numerical methods might still be preferred to analytical ones that are often too cumbersome to obtain. In this paper, we present the Chebyshev pseudospectral or collocation method as an alternative to the hitherto commonly used finite difference schemes (compartmental models) that are based on sufficiently fine equidistant subdivisions of the spatial structure (dendrites or axon). In the Chebyshev method, solutions are approximated by finite Chebyshev series. The solutions have uniform, usually high, numerical accuracy at any spatial point, not only at the original collocation points. Often, truncation errors become negligible, hence, the total error is essentially the rounding error of the computations. Furthermore, quantities involving spatial derivatives, and in particular the axial current, can be computed exactly from the solution, i.e. the membrane potential. Space-dependent parameter distributions (channel densities, non-uniform dendritic geometries), as well as mixed linear boundary conditions can easily be implemented, and can be chosen from the large class of piecewise smooth functions.

1. The existence of a non-negligible steady-state ('window') component of the low threshold, T-type Ca2+current (IT) and an appropriately large ratio of IT to ILeak conductance (i.e. gT/gLeak) have been shown to underlie a novel form of intrinsic bistability that is present in about 15 % of thalamocortical (TC) neurones. 2. In the present experiments, the dynamic clamp technique was used to introduce into mammalian TC neurones in vitro either an artificial, i.e. computer-generated, IT in order to enhance endogenous IT, or an artificial inward ILeak to decrease endogenous ILeak. Using this method, we were able to investigate directly whether the majority of TC neurones appear non-bistable because their intrinsic ionic membrane properties are essentially different (i.e. presence of a negligible IT 'window' component), or simply because they possess a gT or gLeak conductance that is insufficiently large or small, respectively. 3. The validity of the dynamic clamp arrangement and the accuracy of artificial IT were confirmed by (i) recreating the low threshold calcium potential (LTCP) with artificial IT following its block by Ni2+ (0.5-1 mM), and (ii) blocking endogenous LTCPs with an artificial outward IT. 4. Augmentation of endogenous IT by an artificial analog or introduction of an artificial inward ILeak transformed all non-bistable TC neurones to bistable cells that expressed the full array of bistability-mediated behaviours, i.e. input signal amplification, slow oscillatory activity and membrane potential bistability. 5. These results demonstrate the existence of a non-negligible IT 'window' component in all TC neurones and suggest that rather than being a novel group of neurones, bistable cells are merely representative of an interesting region of dynamical modes in the (gT, gLeak) parameter space that may be expressed under certain physiological or pathological conditions by all TC neurones and other types of excitable cells that possess an IT 'window' component with similar biophysical properties.

Recent experiments have produced direct evidence on the existence of various dendritic voltage-gated ion channels, indicating that these neuronal components are not just a passive medium for the propagation of synaptic excitation but a putative source of neuronal excitability that is reflected in the activity patterns occurring on the soma. In order to study possible changes in neuronal excitability when the distribution of dendritic voltage-activated channels is non-uniform, and the dendritic geometry is not necessarily cylindric, we have developed a neuron model that incorporates two voltage-activated currents [I(Na) and I(K)], and in which space-dependent distributions of the system parameters can be treated in a mathematically simple and efficient way. Simulation results with the model showed that both linearly and exponentially tapering geometries led to marked anisotropy of the propagation of excitation, favouring the soma-to-dendrite direction. Exponentially decaying densities of dendritic voltage-activated channels, with appropriate choice of the parameters, induced bistable behaviour between the normal resting state and an intrinsic, sustained oscillation with cylindric as well as linear and exponential tapering dendritic geometry. Bistability could not be evoked when the model was reduced to a space-independent one (point-like soma). These results suggest that both tapering dendritic geometry and inhomogeneous distribution of ion channels may crucially affect the propagation and integration of synaptic potentials, and that changes in dendritic channel densities might underlie pathological electrophysiological activities.

TO investigate the functional role of dendrites of thalamocortical neurones, we have used our one-compartmental model to construct a multi-compartmental model with dendritic regions chosen according to a representative soma-to-dendritic terminal path of an X cell of the cat dorsal lateral geniculate nucleus. The multi-compartmental model with dendritic low-threshold Ca2+ and delayed rectifier K+ channels yields more accurate results than the one-compartmental model when simulating tonic firing and oscillatory activities, and provides a useful means for the study of propagation of excitation on the dendrites.

The mechanism underlying a novel form of input signal amplification and bistability was investigated by intracellular recording in rat and cat thalamocortical (TC) neurones maintained in slices and by computer simulation with a biophysical model of these neurones. In a narrow membrane potential range centred around −60 mV, TC neurones challenged with small (10–50 pA), short (50–200 ms) current steps produced a stereotyped, large amplitude hyperpolarization (> 20 mV) terminated by the burst firing of action potentials, leading to amplification of the duration and amplitude of the input signal, that is hereafter referred to as input signal amplification. In the same voltage range centred around −60 mV, single evoked EPSPs and IPSPs also produced input signal amplification, indicating that this behaviour can be triggered by physiologically relevant stimuli. In addition, a novel, intrinsic, low frequency oscillation, characterized by a peculiar voltage dependence of its frequency and by the presence of plateau potentials on the falling phase of low threshold Ca 2+ potentials, was recorded. Blockade of pure Na + and K + currents by tetrodotoxin (1 μ m ) and Ba 2+ (0.1–2.0 m m ), respectively, did not affect input signal amplification, neither did the presence of excitatory or inhibitory amino acid receptor antagonists in the perfusion medium. A decrease in [Ca 2+ ] 0 (from 2 to 1 m m ) and an increase in [Mg 2+ ] 0 (from 2 to 10 m m ), or the addition of Ni 2+ (2–3 m m ), abolished input signal amplification, while an increase in [Ca 2+ ] 0 (from 2 to 8 m m ) generated this behaviour in neurones where it was absent in control conditions. These results indicate the involvement of the low threshold Ca 2+ current ( I T ) in input signal amplification, since the other Ca 2+ currents of TC neurones are activated at potentials more positive than −40 mV. Blockade of the slow inward mixed cationic current ( I h ) by 4‐( N ‐ethyl‐ N ‐phenylamino)‐1,2‐dimethyl‐6‐(methylamino)‐pyrimidinium chloride (ZD 7288) (100–300 μ m ) did not affect the expression of the large amplitude hyperpolarization, but abolished the subsequent repolarization to the original membrane potential. In this condition, therefore, input signal amplification was replaced by bistable membrane behaviour, where two stable membrane potentials separated by 15–30 mV could be switched between by small current steps. Computer simulation with a model of a TC neurone, which contained only I T , I h , K + leak current ( I Leak ) and those currents responsible for action potentials, accurately reproduced the qualitative and quantitative properties of input signal amplification, bistability and low frequency oscillation, and indicated that these phenomena will occur at some value of the injected DC if, and only if, the ‘window’ component of I T ( I T window) and the leak conductance ( g Leak ) satisfy the relation (d I T Window /d V ) max > g Leak . The physiological implications of these findings for the electroresponsiveness of TC neurones are discussed, and, as I T is widely expressed in the central nervous system, we suggest that ‘window’ I T will markedly affect the integrative properties of many neurones.

THALAMOCORTICAL neurons are capable of displaying complex electrophysiological behaviour, such as lowand high-frequency oscillations, because of their large number of different membrane channels. Recent experimental results have provided direct evidence on the involvement of high-threshold Ca2+ currents (IHVA) in the genesis of the high-frequency oscillations that underlie intermittent action potential firing. Complementing these findings, we now show by means of a biophysical model of a thalamocortical neuron how the interaction of IHVA with a Ca2+-activated K+ current which is also present in these neurons may generate this activity via intracellular Ca2+ processing.

In this paper, we have carried out a theoretical analysis of the recovery process of inactivating currents whose voltage-dependent conductances obey the Hodgkin-Huxley equations. We demonstrate that the recovery process is complex, and, in particular, is non-exponential. Consequently, it cannot be characterized by a single-time constant. Nevertheless, its time-course is completely determined by the properties of the activation and inactivation kinetics at the membrane potential at which the deinactivation of the current takes place. Moreover, we show that the recovery asymptotically approaches an exponential time-course whose time-constant, in turn, is found to be identical to that of the inactivation at the membrane potential of deinactivation. The method commonly used to reconstruct the recovery process can, therefore, provide a way of estimating the inactivation time-constant at membrane potentials where a measurement with the usual voltage-clamp protocol would not be possible. The conclusions of our analysis are discussed with regard to recent theoretical and experimental results.

The voltage-clamp technique is widely employed to obtain data suitable for the reliable estimation of the steady-state and kinetic parameters of inactivating ionic currents in neurones and other excitable cells. Yet, the estimation procedure itself remains a difficult numerical problem, because of the strong non-linear nature of the currents involved. The majority of the numerical methods of parameter estimation makes use of one or another type of non-linear optimization algorithms, and hence is, by nature, iterative. The optimization criterion is based on the maximum likelihood or the least-square error principle and the search for the optimal values takes place in a multi-dimensional parameter space. It is, therefore, prone to be trapped at some local extremum of the parameter space. Moreover, a large number of iterations may be needed to find the optimum using up large amount of computing time. In this paper, we introduce a method that avoids these shortcomings in that it splits up the multi-parameter non-linear fitting problems into a sequence of linear regressions. Furthermore, it uses the value of tp, the time at which the current trace reaches its peak value, to estimate the activation kinetics of the current. Our approach also guarantees that the estimates will be sufficiently close to the ‘real’ values, provided the quality of the experimental records is satisfactory. In order to test our method, we used kinetic and steady-state properties of the following three currents as identified in earlier experiments: the low-threshold Ca2+ current, IT, and the K+ currents, IA and IK2. Gaussian noise of constant variance was added to the simulated current traces. The method was also tested on experimental traces of IT.

A large amount of experimental data has provided direct evidence that thalamocortical (TC) cells in vitro are capable of different types of intrinsic oscillatory activities, that is, oscillations that occur in the absence of sensory and cortical inputs and of intrathalamic afferents, that is, from interneurons and cells of the nucleus reticularis thalami. This chapter describes these types of oscillation, analyzes their underlying ionic currents, and presents a biophysical model of a TC cell that accurately reproduces the intrinsic activity patterns and the transitions among these different states. In addition, the chapter also discusses how rhythmic and randomly occurring fast synaptic potentials—EPSPs and IPSPs—modulate these intrinsic oscillations, and stresses that the behavior of TC cells in both physiological and pathological conditions can be fully understood only by taking into account that they are pacemaker neurons whose activity is modulated by synaptic inputs.