Movement-related phase locking in the delta-theta frequency band.

Movements result from a complex interplay of multiple brain regions. These regions are assembled into distinct functional networks depending on the specific properties of the action. However, the nature and details of the dynamics of this complex assembly process are unknown. In this study, we sought to identify key markers of the neural processes underlying the preparation and execution of motor actions that always occur irrespective of differences in movement initiation, hence the specific neural processes and functional networks involved. To this end, EEG activity was continuously recorded from 18 right-handed healthy participants while they performed a simple motor task consisting of button presses with the left or right index finger. The movement was performed either in response to a visual cue or at a self-chosen, i.e., non-cued point in time. Despite these substantial differences in movement initiation, dynamic properties of the EEG signals common to both conditions could be identified using time-frequency and phase locking analysis of the EEG data. In both conditions, a significant phase locking effect was observed that started prior to the movement onset in the δ-θ frequency band (2-7Hz), and that was strongest at the electrodes nearest to the contralateral motor region (M1). This phase locking effect did not have a counterpart in the corresponding power spectra (i.e., amplitudes), or in the event-related potentials. Our finding suggests that phase locking in the δ-θ frequency band is a ubiquitous movement-related signal independent of how the actual movement has been initiated. We therefore suggest that phase-locked neural oscillations in the motor cortex are a prerequisite for the preparation and execution of motor actions.

A three-leg model producing tetrapod and tripod coordination patterns of ipsilateral legs in the stick insect.

Insect locomotion requires the precise coordination of the movement of all six legs. Detailed investigations have revealed that the movement of the legs is controlled by local dedicated neuronal networks, which interact to produce walking of the animal. The stick insect is well suited to experimental investigations aimed at understanding the mechanisms of insect locomotion. Beside the experimental approach, models have also been constructed to elucidate those mechanisms. Here, we describe a model that replicates both the tetrapod and tripod coordination pattern of three ipsilateral legs. The model is based on an earlier insect leg model, which includes the three main leg joints, three antagonistic muscle pairs, and their local neuronal control networks. These networks are coupled via angular signals to establish intraleg coordination of the three neuromuscular systems during locomotion. In the present three-leg model, we coupled three such leg models, representing front, middle, and hind leg, in this way. The coupling was between the levator-depressor local control networks of the three legs. The model could successfully simulate tetrapod and tripod coordination patterns, as well as the transition between them. The simulations showed that for the interleg coordination during tripod, the position signals of the levator-depressor neuromuscular systems sent between the legs were sufficient, while in tetrapod, additional information on the angular velocities in the same system was necessary, and together with the position information also sufficient. We therefore suggest that, during stepping, the connections between the levator-depressor neuromuscular systems of the different legs are of primary importance.

A network model comprising 4 segmental, interconnected ganglia, and its application to simulate multi-legged locomotion in crustaceans.

Inter-segmental coordination is crucial for the locomotion of animals. Arthropods show high variability of leg numbers, from 6 in insects up to 750 legs in millipedes. Despite this fact, the anatomical and functional organization of their nervous systems show basic similarities. The main similarities are the segmental organization, and the way the function of the segmental units is coordinated. We set out to construct a model that could describe locomotion (walking) in animals with more than 6 legs, as well as in 6-legged animals (insects). To this end, we extended a network model by Daun-Gruhn and Tóth (Journal of Computational Neuroscience, doi: 10.1007/s10827-010-0300-1 , 2011). This model describes inter-segmental coordination of the ipsilateral legs in the stick insect during walking. Including an additional segment (local network) into the original model, we could simulate coordination patterns that occur in animals walking on eight legs (e.g., crayfish). We could improve the model by modifying its original cyclic connection topology. In all model variants, the phase relations between the afferent segmental excitatory sensory signals and the oscillatory activity of the segmental networks played a crucial role. Our results stress the importance of this sensory input on the generation of different stable coordination patterns. The simulations confirmed that using the modified connection topology, the flexibility of the model behaviour increased, meaning that changing a single phase parameter, i.e., gating properties of just one afferent sensory signal was sufficient to reproduce all coordination patterns seen in the experiments.

A neuromechanical model for the neuronal basis of curve walking in the stick insect.

The coordination of the movement of single and multiple limbs is essential for the generation of locomotion. Movement about single joints and the resulting stepping patterns are usually generated by the activity of antagonistic muscle pairs. In the stick insect, the three major muscle pairs of a leg are the protractor and retractor coxae, the levator and depressor trochanteris, and the flexor and extensor tibiae. The protractor and retractor move the coxa, and thereby the leg, forward and backward. The levator and depressor move the femur up and down. The flexor flexes, and the extensor extends the tibia about the femur-tibia joint. The underlying neuronal mechanisms for a forward stepping middle leg have been thoroughly investigated in experimental and theoretical studies. However, the details of the neuronal and mechanical mechanisms driving a stepping single leg in situations other than forward walking remain largely unknown. Here, we present a neuromechanical model of the coupled three joint control system of the stick insect's middle leg. The model can generate forward, backward, or sideward stepping. Switching between them is achieved by changing only a few central signals controlling the neuromechanical model. In kinematic simulations, we are able to generate curve walking with two different mechanisms. In the first, the inner middle leg is switched from forward to sideward and in the second to backward stepping. Both are observed in the behaving animal, and in the model and animal alike, backward stepping of the inner middle leg produces tighter turns than sideward stepping.

A neuromechanical model explaining forward and backward stepping in the stick insect.

The mechanism underlying the generation of stepping has been the object of intensive studies. Stepping involves the coordinated movement of different leg joints and is, in the case of insects, produced by antagonistic muscle pairs. In the stick insect, the coordinated actions of three such antagonistic muscle pairs produce leg movements and determine the stepping pattern of the limb. The activity of the muscles is controlled by the nervous system as a whole and more specifically by local neuronal networks for each muscle pair. While many basic properties of these control mechanisms have been uncovered, some important details of their interactions in various physiological conditions have so far remained unknown. In this study, we present a neuromechanical model of the coupled protractor-retractor and levator-depressor neuromuscular systems and use it to elucidate details of their coordinated actions during forward and backward walking. The switch from protraction to retraction is evoked at a critical angle of the femur during downward movement. This angle represents a sensory input that integrates load, motion, and ground contact. Using the model, we can make detailed suggestions as to how rhythmic stepping might be generated by the central pattern generators of the local neuronal networks, how this activity might be transmitted to the corresponding motoneurons, and how the latter might control the activity of the related muscles. The entirety of these processes yields the coordinated interaction between neuronal and mechanical parts of the system. Moreover, we put forward a mechanism by which motoneuron activity could be modified by a premotor network and suggest that this mechanism might serve as a basis for fast adaptive behavior, like switches between forward and backward stepping, which occur, for example, during curve walking, and especially sharp turning, of insects.

The properties of reticular thalamic neuron GABA(A) IPSCs of absence epilepsy rats lead to enhanced network excitability.

Both human investigations and studies in animal models have suggested that abnormalities in GABA(A) receptor function have a potential role in the pathophysiology of absence seizures. Recently we showed that, prior to seizure onset, GABA(A) IPSCs in thalamic reticular (NRT) neurons of genetic absence epilepsy rats from Strasbourg (GAERS) had a 25% larger amplitude, a 40% faster decay and a 45% smaller paired-pulse depression than those of nonepileptic control (NEC) rats. By means of a novel mathematical description, the properties of both GAERS and NEC GABAergic synapses can be mimicked. These model synapses were then used in an NRT network model in order to investigate their potential impact on the neuronal firing patterns. Compared to NEC, GAERS NRT neurons show an overall increase in excitability and a higher frequency and regularity of firing in response to periodic input signals. Moreover, in response to randomly distributed stimuli, the GAERS but not the NEC model produces resonance between 7 and 9 Hz, the frequency range of spike-wave discharges in GAERS. The implications of these results for the epileptogenesis of absence seizures are discussed.

Estimation of the activation and kinetic properties of I(Na) and I(K) from the time course of the action potential.

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).

Backpropagation of the delta oscillation and the retinal excitatory postsynaptic potential in a multi-compartment model of thalamocortical neurons.

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.

All thalamocortical neurones possess a T-type Ca2+ 'window' current that enables the expression of bistability-mediated activities.

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.

Solution of the nerve cable equation using Chebyshev approximations.

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.

Analysis and biophysical interpretation of bistable behaviour in thalamocortical neurons.

Thalamocortical neurons display a wide spectrum of activity patterns that are the expressions of the non-linear interactions between the various voltage-gated ion channels. Here, we show how bistable behaviour can emerge in these neurons, and how it is brought about by the steady-state residual ("window") component of IT, the low-threshold Ca2+ current. In particular, we present results that describe the dependence of bistability on two system parameters: the injected current and the leakage conductance. In addition, we provide a biophysical interpretation of these results by means of the properties of the electrical circuit representing the neuron membrane.

Effects of tapering geometry and inhomogeneous ion channel distribution in a neuron model.

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.

Simulation of intermittent action potential firing in thalamocortical neurons.

Thalamocortical neurons are capable of displaying complex electrophysiological behaviour, such as low- and 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 (I(HVA)) 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 I(HVA) with a Ca2+-activated K+ current which is also present in these neurons may generate this activity via intracellular Ca2+ processing.

Model of a thalamocortical neurone with dendritic voltage-gated ion channels.

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 'window' component of the low threshold Ca2+ current produces input signal amplification and bistability in cat and rat thalamocortical neurones.

1. 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. 2. 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. 3. 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 Ca2+ potentials, was recorded. 4. Blockade of pure Na+ and K+ currents by tetrodotoxin (1 microM) and Ba2+ (0.1-2.0 mM), respectively, did not affect input signal amplification, neither did the presence of excitatory or inhibitory amino acid receptor antagonists in the perfusion medium. 5. A decrease in [Ca2+]o (from 2 to 1 mM) and an increase in [Mg2+]o (from 2 to 10 mM), or the addition of Ni2+ (2-3 mM), abolished input signal amplification, while an increase in [Ca2+]o (from 2 to 8 mM) generated this behaviour in neurones where it was absent in control conditions. These results indicate the involvement of the low threshold Ca2+ current (IT) in input signal amplification, since the other Ca2+ currents of TC neurones are activated at potentials more positive than -40 mV. 6. Blockade of the slow inward mixed cationic current (Ih) by 4-(N-ethyl-N-phenylamino)-1,2-dimethyl-6-(methylamino)-pyrimidinium++ + chloride (ZD 7288)(100-300 microM) 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. 7. Computer simulation with a model of a TC neurone, which contained only IT, Ih, K+ leak current (ILeak) 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 IT (IT,Window) and the leak conductance (gLeak) satisfy the relation (dIT,Window/dV)max > gLeak. 8. The physiological implications of these findings for the electroresponsiveness of TC neurones are discussed, and, as IT is widely expressed in the central nervous system, we suggest that 'window' IT will markedly affect the integrative properties of many neurones.

Model of a thalamocortical neurone with dendritic voltage-gated ion channels.

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 theoretical basis for non-exponential time-course of the recovery from inactivation of Hodgkin-Huxley-type ionic currents.

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.

A numerical procedure to estimate kinetic and steady-state characteristics of inactivating ionic currents.

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 model of ion channel kinetics using deterministic chaotic rather than stochastic processes.

Models of ion channel kinetics have previously assumed that the switching between the open and closed states is an intrinsically random process. Here, we present an alternative model based on a deterministic process. This model is a piecewise linear iterated map. We calculate the dwell time distributions, autocorrelation function, and power spectrum of this map. We also explore non-linear generalizations of this map. The chaotic nature of our model implies that its long-term behavior mimics the stochastic properties of a random process. In particular, the linear map produces an exponential probability distribution of dwell times in the open and closed states, the same as that produced by the two-state, closed in equilibrium open, Markov model. We show how deterministic and random models can be distinguished by their different phase space portraits. A test of some experimental data seems to favor the deterministic model, but further experimental evidence is needed for an unequivocal decision.