A biophysical model of nonlinear dynamics underlying plateau potentials and calcium spikes in purkinje cell dendrites

Purkinje cell models: past, present and future

The investigation of the dynamics of Purkinje cell (PC) activity is crucial to unravel the role of the cerebellum in motor control, learning and cognitive processes. Within the cerebellar cortex (CC), these neurons receive all the incoming sensory and motor information, transform it and generate the entire cerebellar output. The relatively homogenous and repetitive structure of the CC, common to all vertebrate species, suggests a single computation mechanism shared across all PCs. While PC models have been developed since the 70's, a comprehensive review of contemporary models is currently lacking. Here, we provide an overview of PC models, ranging from the ones focused on single cell intracellular PC dynamics, through complex models which include synaptic and extrasynaptic inputs. We review how PC models can reproduce physiological activity of the neuron, including firing patterns, current and multistable dynamics, plateau potentials, calcium signaling, intrinsic and synaptic plasticity and input/output computations. We consider models focusing both on somatic and on dendritic computations. Our review provides a critical performance analysis of PC models with respect to known physiological data. We expect our synthesis to be useful in guiding future development of computational models that capture real-life PC dynamics in the context of cerebellar computations.

A model of on/off transitions in neurons of the deep cerebellar nuclei: deciphering the underlying ionic mechanisms

The neurons of the deep cerebellar nuclei (DCNn) represent the main functional link between the cerebellar cortex and the rest of the central nervous system. Therefore, understanding the electrophysiological properties of DCNn is of fundamental importance to understand the overall functioning of the cerebellum. Experimental data suggest that DCNn can reversibly switch between two states: the firing of spikes (F state) and a stable depolarized state (SD state). We introduce a new biophysical model of the DCNn membrane electro-responsiveness to investigate how the interplay between the documented conductances identified in DCNn give rise to these states. In the model, the F state emerges as an isola of limit cycles, i.e. a closed loop of periodic solutions disconnected from the branch of SD fixed points. This bifurcation structure endows the model with the ability to reproduce the [Formula: see text] transition triggered by hyperpolarizing current pulses. The model also reproduces the [Formula: see text] transition induced by blocking Ca currents and ascribes this transition to the blocking of the high-threshold Ca current. The model suggests that intracellular current injections can trigger fully reversible [Formula: see text] transitions. Investigation of low-dimension reduced models suggests that the voltage-dependent Na current is prominent for these dynamical features. Finally, simulations of the model suggest that physiological synaptic inputs may trigger [Formula: see text] transitions. These transitions could explain the puzzling observation of positively correlated activities of connected Purkinje cells and DCNn despite the former inhibit the latter.

The role of relative membrane capacitance and time delay in cerebellar Purkinje cells

The membrane capacitance of a neuron can influence the synaptic efficacy and the speed of electrical signal propagation. Exploring the role of membrane capacitance will help facilitate a deeper understanding of the electrical properties of neurons. Thus, in this paper, we investigated the neuronal firing behaviors of a two-compartment model in Purkinje cells. We evaluated the influence of membrane capacitance under two different circumstances: in the absence of time delay and in the presence of time delay. Firstly, we separately studied the influence of somatic membrane capacitance Cs and dendritic membrane capacitance Cd on neuronal firing patterns. Through numerical simulation, we observed that they had two different types of period-adding scenarios, i.e. with and without chaotic bursting. Secondly, our results indicated that when the time delay was included in the model, periodic motions were more inclined to be destroyed, while at the same time, corresponding new chaotic motions were induced. These findings suggested that membrane capacitance and time delay play a pivotal functional role in modulating dynamical firing properties of neurons, especially aspects which lead to behaviors which result in changes to bursting patterns.

Conditional Bistability, a Generic Cellular Mnemonic Mechanism for Robust and Flexible Working Memory Computations

Persistent neural activity, the substrate of working memory, is thought to emerge from synaptic reverberation within recurrent networks. However, reverberation models do not robustly explain the fundamental dynamics of persistent activity, including high-spiking irregularity, large intertrial variability, and state transitions. While cellular bistability may contribute to persistent activity, its rigidity appears incompatible with persistent activity labile characteristics. Here, we unravel in a cellular model a form of spike-mediated conditional bistability that is robust and generic. and provides a rich repertoire of mnemonic computations. Under asynchronous synaptic inputs of the awakened state, conditional bistability generates spiking/bursting episodes, accounting for the irregularity, variability, and state transitions characterizing persistent activity. This mechanism has likely been overlooked because of the subthreshold input it requires, and we predict how to assess it experimentally. Our results suggest a reexamination of the role of intrinsic properties in the collective network dynamics responsible for flexible working memory.SIGNIFICANCE STATEMENT This study unravels a novel form of intrinsic neuronal property: conditional bistability. We show that, thanks to its conditional character, conditional bistability favors the emergence of flexible and robust forms of persistent activity in PFC neural networks, in opposition to previously studied classical forms of absolute bistability. Specifically, we demonstrate for the first time that conditional bistability (1) is a generic biophysical spike-dependent mechanism of layer V pyramidal neurons in the PFC and that (2) it accounts for essential neurodynamical features for the organization and flexibility of PFC persistent activity (the large irregularity and intertrial variability of the discharge and its organization under discrete stable states), which remain unexplained in a robust fashion by current models.

A simple transfer function for nonlinear dendritic integration

Relatively recent advances in patch clamp recordings and iontophoresis have enabled unprecedented study of neuronal post-synaptic integration ("dendritic integration"). Findings support a separate layer of integration in the dendritic branches before potentials reach the cell's soma. While integration between branches obeys previous linear assumptions, proximal inputs within a branch produce threshold nonlinearity, which some authors have likened to the sigmoid function. Here we show the implausibility of a sigmoidal relation and present a more realistic transfer function in both an elegant artificial form and a biophysically derived form that further considers input locations along the dendritic arbor. As the distance between input locations determines their ability to produce nonlinear interactions, models incorporating dendritic topology are essential to understanding the computational power afforded by these early stages of integration. We use the biophysical transfer function to emulate empirical data using biophysical parameters and describe the conditions under which the artificial and biophysically derived forms are equivalent.

Storage of correlated patterns in standard and bistable Purkinje cell models

The cerebellum has long been considered to undergo supervised learning, with climbing fibers acting as a 'teaching' or 'error' signal. Purkinje cells (PCs), the sole output of the cerebellar cortex, have been considered as analogs of perceptrons storing input/output associations. In support of this hypothesis, a recent study found that the distribution of synaptic weights of a perceptron at maximal capacity is in striking agreement with experimental data in adult rats. However, the calculation was performed using random uncorrelated inputs and outputs. This is a clearly unrealistic assumption since sensory inputs and motor outputs carry a substantial degree of temporal correlations. In this paper, we consider a binary output neuron with a large number of inputs, which is required to store associations between temporally correlated sequences of binary inputs and outputs, modelled as Markov chains. Storage capacity is found to increase with both input and output correlations, and diverges in the limit where both go to unity. We also investigate the capacity of a bistable output unit, since PCs have been shown to be bistable in some experimental conditions. Bistability is shown to enhance storage capacity whenever the output correlation is stronger than the input correlation. Distribution of synaptic weights at maximal capacity is shown to be independent on correlations, and is also unaffected by the presence of bistability.

Potassium currents of olfactory bulb juxtaglomerular cells: characterization, simulation, and implications for plateau potential firing

Odor identity is encoded by the activity of olfactory bulb glomeruli, which receive primary sensory input and transfer it to projection neurons. Juxtaglomerular cells (JGCs) may influence glomerular processing via firing of long lasting plateau potentials. Though inward currents have been investigated, little is known regarding potassium current contribution to JGC plateau potentials. We pursued study of these currents, with the overarching goal of creating components for a computational model of JGC plateau potential firing. In conditions minimizing calcium-activated potassium current (I(K(Ca))), we used whole cell voltage clamp and in vitro slice preparations to characterize three potassium currents in rat JGCs. The prominent component I(kt1) displayed rapid kinetics (τ(10%-90% rise), 0.6-2 ms; τ(inactivation), 5-10 ms) and was blocked by high concentration 4-aminopyridine (4-AP) (5 mM) and tetramethylammonium (TEA) (40 mM). It had half maximal activation at -10 mV (V(½)max) and little inactivation at rest. I(kt2), with slower kinetics (τ(10%-90% rise), 11-15 ms; τ(inactivation), 100-300 ms), was blocked by low concentration 4-AP (0.5 mM) and TEA (5 mM). The V(½)max was 0 mV and inactivation was also minimal at rest. Sustained current I(kt3) showed sensitivity to low concentration 4-AP and TEA, and had V(½)max of +10 mV. Further experiments, in conditions of physiologic calcium buffering, suggested that I(K(Ca)) contributed to I(kt3) with minimal effect on plateau potential evolution. We transformed these characterizations into Hodgkin-Huxley models that robustly mimicked experimental data. Further simulation demonstrated that I(kt1) would be most efficiently activated by plateau potential waveforms, predicting a critical role in shaping JGC firing. These studies demonstrated that JGCs possess a unique potassium current profile, with delayed rectifier (I(kt3)), atypical A-current (I(kt1)), and D-current (I(kt2)) in accordance with known expression patterns in olfactory bulb (OB) glomeruli. Our simulations also provide an initial framework for more integrative models of JGC plateau potential firing.

Calcium currents of olfactory bulb juxtaglomerular cells: profile and multiple conductance plateau potential simulation

The olfactory glomerulus is the locus of information transfer between olfactory sensory neurons and output neurons of the olfactory bulb. Juxtaglomerular cells (JGCs) may influence intraglomerular processing by firing plateau potentials that support multiple spikes. It is unclear what inward currents mediate this firing pattern. In previous work, we characterized potassium currents of JGCs. We focus here on the inward currents using whole cell current clamp and voltage recording in a rat in vitro slice preparation, as well as computer simulation. We first showed that sodium current was not required to mediate plateau potentials. Voltage clamp characterization of calcium current (I(Ca)) determined that I(Ca) consisted of a slow activating, rapidly inactivating (τ(10%-90% rise) 6-8 ms, τ(inactivation) 38-77 ms) component I(cat1), similar to T-type currents, and a sustained (τ(inactivation)>>500 ms) component I(cat2), likely composed of L-type and P/Q-type currents. We used computer simulation to test their roles in plateau potential firing. We robustly modeled I(cat1) and I(cat2) to Hodgkin-Huxley schemes (m(3)h and m(2), respectively) and simulated a JGC plateau potential with six conductances: calcium currents as above, potassium currents from our prior study (A-type I(kt1), D-type I(kt2), delayed rectifier I(kt3)), and a fast sodium current (I(Na)). We demonstrated that I(cat1) was required for mediating the plateau potential, unlike I(Na) and I(cat2), and its τ(inactivation) determined plateau duration. We also found that I(kt1) dictated plateau potential shape more than I(kt2) and I(kt3). The influence of these two transient and opposing conductances suggests a unique mechanism of plateau potential physiology.

An elementary model of torus canards

We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.

Dendritic signals command firing dynamics in a mathematical model of cerebellar Purkinje cells

Dendrites of cerebellar Purkinje cells (PCs) respond to brief excitations from parallel fibers with lasting plateau depolarizations. It is unknown whether these plateaus are local events that boost the synaptic signals or they propagate to the soma and directly take part in setting the cell firing dynamics. To address this issue, we analyzed a likely mechanism underlying plateaus in three representations of a reconstructed PC with increasing complexity. Analysis in an infinite cable suggests that Ca plateaus triggered by direct excitatory inputs from parallel fibers and their mirror signals, valleys (putatively triggered by the local feed forward inhibitory network), cannot propagate. However, simulations of the model in electrotonic equivalent cables prove that Ca plateaus (resp. valleys) are conducted over the entire cell with velocities typical of passive events once they are triggered by threshold synaptic inputs that turn the membrane current inward (resp. outward) over the whole cell surface. Bifurcation analysis of the model in equivalent cables, and simulations in a fully reconstructed PC both indicate that dendritic Ca plateaus and valleys, respectively, command epochs of firing and silencing of PCs.

A biologically plausible model of time-scale invariant interval timing

The temporal durations between events often exert a strong influence over behavior. The details of this influence have been extensively characterized in behavioral experiments in different animal species. A remarkable feature of the data collected in these experiments is that they are often time-scale invariant. This means that response measurements obtained under intervals of different durations coincide when plotted as functions of relative time. Here we describe a biologically plausible model of an interval timing device and show that it is consistent with time-scale invariant behavior over a substantial range of interval durations. The model consists of a set of bistable units that switch from one state to the other at random times. We first use an abstract formulation of the model to derive exact expressions for some key quantities and to demonstrate time-scale invariance for any range of interval durations. We then show how the model could be implemented in the nervous system through a generic and biologically plausible mechanism. In particular, we show that any system that can display noise-driven transitions from one stable state to another can be used to implement the timing device. Our work demonstrates that a biologically plausible model can qualitatively account for a large body of data and thus provides a link between the biology and behavior of interval timing.

Suppressing the excitability of spinal motoneurons by extracellularly applied electrical fields: insights from computer simulations

The effect of extracellularly applied electrical fields on neuronal excitability and firing behavior is attributed to the interaction between neuronal morphology and the spatial distribution and level of differential polarization induced by the applied field in different elements of the neuron. The presence of voltage-gated ion channels that mediate persistent inward currents (PICs) on the dendrites of spinal motoneurons enhances the influence of electrical fields on the motoneuronal firing behavior. The goal of the present study was to investigate, with a realistic motoneuron computer model, the effects of extracellularly applied electrical fields on the excitability of spinal motoneurons with the aim of reducing the increased motoneuronal excitability after spinal cord injury (SCI). Our results suggest that electrical fields could suppress the excitability of motoneurons and reduce their firing rate significantly by modulating the magnitude of their dendritic PIC. This effect was achieved at different field directions, intensities, and polarities. The reduction in motoneuronal firing rate resulted from the reduction in the magnitude of the dendritic PIC reaching the soma by the effect of the applied electrical field. This reduction in PIC was attributed to the dendritic field-induced differential polarization and the nonlinear current-voltage relationship of the dendritic PIC-mediating channels. Because of the location of the motoneuronal somata and initial segment with respect to the dendrites, these structures were minimally polarized by the applied field compared with the extended dendrites. In conclusion, electrical fields could be used for suppressing the hyperexcitability of spinal motoneurons after SCI and reducing the level of spasticity.

Transformation of the kinematic characteristics of a precise movement after a change in a spatial task

The central mechanism of motor programming was studied using a model of precise horizontal flexion of the arm at the elbow joint. Training was performed in the dark to ensure that formation of the motor program was based predominantly on the use of proprioceptive afferentation. The target was not demonstrated before training: subjects determined the angle of arm flexion during training, the moment at which the target position was reached being identified by a brief LED flash. Subjects had to perform the movement as quickly and accurately as possible. The amplitude, speed, and accuracy of the movement were measured in real time. The ten subjects were divided into two groups for initial training to precise movements, using two different protocols: flexion of the elbow to 70 degrees and to 55 degrees . At the second stage of the experiment, each subject's initial target position was suddenly changed (from 70 degrees to 55 degrees and vice versa). Training was continued until a stable accuracy in the new conditions was achieved (with errors of no more than 5% of the specified amplitude). The nature of the transformation in the kinematics of the precise movement in response to the change in the single task parameter illuminated the fundamental principle of organization of the supraspinal motor command for movements of this type. For both specified flexion amplitudes, the ratio between the acceleration and deceleration phases of the movement were identical during the period of skill fixation. On average, 70% of the total amplitude of flexion was accounted for by the acceleration phase and 30% by the deceleration phase. Adaptation of the precise movement to changes in the specified horizontal elbow flexion angle (i.e., re-achievement of the required movement accuracy in the changed conditions) during rearrangement was completed by inversion of these values. According to the results of previous studies, the most informative measure for analysis of the dynamics of the controlling central command was the acceleration of the movement. In terms of current concepts of the mechanism of motor control, the acceleration plateau can be regarded as mirroring long-term depression--the voltage plateau in Purkinje cells and motoneurons. Data processing involved calculation of the integral acceleration in both phases of the movement in relation to the angle of flexion (phase plots). These data underlie our understanding of the mechanism of transformation of movement kinematics responsible for the formation of a new central command.

Firing dynamics of cerebellar purkinje cells

Knowledge of intrinsic neuronal firing dynamics is a critical first step to establishing an accurate biophysical model of any neuron. In this study we examined cerebellar Purkinje cells to determine the bifurcations likely to underlie firing dynamics within a biophysically realistic and experimentally supported model. We show that Purkinje cell dynamics are consistent with a system undergoing a saddle-node bifurcation of fixed points in the transition from rest to firing and a saddle homoclinic bifurcation from firing to rest. Our analyses account for numerous observed Purkinje cell firing properties that include bistability, plateau potentials, specific aspects of the frequency-current (F-I) relationship, first spike latency, and the ability for climbing fiber input to induce state transitions in the bistable regime. We also experimentally confirm new properties predicted from our model and analysis that include the presence of a depolarizing afterpotential (DAP), the ability to fire at low frequencies (<50 Hz) and with a high gain in the F-I relationship, and a bistable region limited to low-frequency firing. Purkinje cell dynamics, including bistability, prove to arise from numerous biophysical factors that include the DAP, fast refractory dynamics, and a long membrane time constant. A hyperpolarizing activated cation current (I(H)) is shown not to be directly involved in establishing bistable dynamics but rather reduces the range for bistability. A combined electrophysiological and modeling approach thus accounts for several properties of Purkinje cells, providing a firm basis from which to assess Purkinje cell output patterns.

Agents of the mind

The higher order circuitry of the brain is comprised of a large-scale network of cerebral cortical areas that are individually regulated by loops through subcortical structures, particularly through the basal ganglia and cerebellum. These subcortical loops have powerful computational architectures. Using, as an example, the relatively well-understood processing that occurs in the cortical/basal ganglionic/cerebellar distributed processing module that generates voluntary motor commands, I postulate that a network of analogous agents is an appropriate framework for exploring the dynamics of the mind.

A Purkinje cell specific GoLoco domain protein, L7/Pcp-2, modulates receptor-mediated inhibition of Cav2.1 Ca2+ channels in a dose-dependent manner

L7/Pcp-2 is a GoLoco domain protein encoded by a Purkinje cell dendritic mRNA. Although biochemical interactions of GoLoco proteins with Galpha(o) and Galpha(i) are well documented, little is known about effector function modulation resulting from these interactions. The P-type Ca2+ channels might be physiological effectors of L7 because (1) they are the major voltage-dependent Ca2+ channels (VDCC) that modulate Purkinje cell output and (2) they are regulated by G(i/o) proteins. As a first step towards validating this hypothesis and to further understand the possible physiological effect of L7 protein and its two isoforms, we have coexpressed Ca(v)2.1 channels and kappa-opioid receptors (KORs) with varying amounts of L7A or L7B in Xenopus oocytes and measured ionic currents by two-electrode voltage clamping. Without receptor activation L7 did not alter the Ca2+ channel activity. With tonic and weak activation of the receptors, however, the Ca2+ channels were inhibited by 40-50%. This inhibition was enhanced by low, but dampened by high, expression levels of L7A and L7B and differences were observed between the two isoforms. The enhancing effect of L7 was occluded by overexpression of Gbetagamma, whereas the disinhibition was antagonized by overexpression of Galpha(o). We propose that L7 differentially affects the Galpha and Gbetagamma arms of receptor-induced G(i/o) signaling in a concentration-dependent manner, through which it increases the dynamic range of regulation of P/Q-type Ca2+ channels by G(i/o) protein-coupled receptors. This provides a framework for designing further experiments to determine how dendritic local fluctuations in L7 protein levels might influence signal processing in Purkinje cells.

Bistability of cerebellar Purkinje cells modulated by sensory stimulation

A persistent change in neuronal activity after brief stimuli is a common feature of many neuronal microcircuits. This persistent activity can be sustained by ongoing reverberant network activity or by the intrinsic biophysical properties of individual cells. Here we demonstrate that rat and guinea pig cerebellar Purkinje cells in vivo show bistability of membrane potential and spike output on the time scale of seconds. The transition between membrane potential states can be bidirectionally triggered by the same brief current pulses. We also show that sensory activation of the climbing fiber input can switch Purkinje cells between the two states. The intrinsic nature of Purkinje cell bistability and its control by sensory input can be explained by a simple biophysical model. Purkinje cell bistability may have a key role in the short-term processing and storage of sensory information in the cerebellar cortex.

Background synaptic activity modulates the response of a modeled purkinje cell to paired afferent input

We studied the possible effects of background excitatory and inhibitory synaptic activity on Purkinje cell responses to paired pulse inputs using a morphologically and physiologically detailed compartmental model. Paired-pulse inputs were provided via synapses associated with the ascending segment of the granule cell axon in the presence of variable amounts of background synaptic input provided by parallel fibers and inhibitory interneurons. The results predict the amplitude and dynamics of the somatic and dendritic responses to paired ascending segment inputs are modulated by the level of background synaptic activity, the basal somatic firing rate of the Purkinje cell model does not indicate the type or degree of modulation, and the modulatory effects are mediated through dendritic calcium and calcium-activated potassium currents. These results have important consequences for the functional organization of cerebellar cortex as well as the more general issue of the modulatory as distinct from driving effects of synapses now also being considered in other regions of the mammalian brain including the cerebral cortex.

Eye movements of the murine P/Q calcium channel mutant rocker, and the impact of aging

Mutations in the gene encoding the ion pore of the P/Q voltage-activated calcium channel (CACNA1A) are predicted to alter synaptic transmission and dendritic excitability within cerebellar granule and Purkinje cells. Determining the relationships between these alterations, neuronal activity, and behavior may yield insight into the relationship between neuronal intrinsic properties and signal processing within the ocular motor system. Toward this end, we compared ocular motor performance in the CACNA1A mutant rocker and C57BL/6 controls. Average vertical eye position was abnormally elevated in the mutants, a finding that may be analogous to downbeat nystagmus seen in human cerebellar disorders. Fast phases of vestibular nystagmus were slowed by approximately 18% of control values. The angular vestibuloocular reflex (VOR) in darkness and light (visual VOR, or VVOR), assessed at 0.1-1.6 Hz, exhibited subnormal gains at the highest stimulus frequencies and increased phase leads at the lowest stimulus frequencies. Horizontal optokinetic responses to constant velocity drum rotation of +/-2.5-40 degrees/s exhibited minimally reduced gains. Attempts to increase VOR gain by concomitant optokinetic and vestibular stimulation were confounded by the tendency of the mice to habituate to repetitive vestibular stimulation, but attempts to induce coupling of vertical eye movements to horizontal vestibular stimulation (cross-axis adaptation) generated rapid plastic changes in controls and little effect in mutants. With the notable exceptions of the vertical elevation and optokinetic gains, the ocular motor abnormalities were stable over a broad range of animal age, a result compatible with the abnormalities arising as direct consequences of the inborn alteration in calcium channel biophysics.