Period doubling of calcium spike firing in a model of a Purkinje cell dendrite

Recordings from cerebellar Purkinje cell dendrites have revealed that in response to sustained current injection, the cell firing pattern can move from tonic firing of Ca²⁺ spikes to doublet firing and even to quadruplet firing or more complex firing. These firing patterns are not modified substantially if Na⁺ currents are blocked. We show that the experimental results can be viewed as a slow transition of the neuronal dynamics through a period-doubling bifurcation. To further support this conclusion and to understand the underlying mechanism that leads to doublet firing, we develop and study a simple, one-compartment model of Purkinje cell dendrite. The neuron can also exhibit quadruplet and chaotic firing patterns that are similar to the firing patterns that some of the Purkinje cells exhibit experimentally. The effects of parameters such as temperature, applied current, and potassium reversal potential in the model resemble their effects in experiments. The model dynamics involve three time scales. Ca²⁺- dependent K⁺ currents, with intermediate time scales, are responsible for the appearance of doublet firing, whereas a very slow hyperpolarizing current transfers the neuron from tonic to doublet firing. We use the fast-slow analysis to separate the effects of the three time scales. Fast-slow analysis of the neuronal dynamics, with the activation variable of the very slow, hyperpolarizing current considered as a parameter, reveals that the transitions occurs via a cascade of period-doubling bifurcations of the fast and intermediate subsystem as this slow variable increases. We carry out another analysis, with the Ca²⁺ concentration considered as a parameter, to investigate the conditions for the generation of doublet firing in systems with one effective variable with intermediate time scale, in which the rest state of the fast subsystem is terminated by a saddle-node bifurcation. We find that the scenario of period doubling in these systems can occur only if (1) the time scale of the intermediate variable (here, the decay rate of the calcium concentration) is slow enough in comparison with the interspike interval of the tonic firing at the transition but is not too slow and (2) there is a bistability of the fast subsystem of the spike-generating variables.     

Noise-induced neural impulses

The firing pattern of neural pulses often show the following features: the shapes of individual pulses are nearly identical and frequency independent; the firing frequency can vary over a broad range; the time period between pulses shows a stochastic scatter. This behaviour cannot be understood on the basis of a deterministic non-linear dynamic process, e.g. the Bonhoeffer-van der Pol model. We demonstrate in this paper that a noise term added to the Bonhoeffer-van der Pol model can reproduce the firing patterns of neurons very well. For this purpose we have considered the Fokker-Planck equation corresponding to the stochastic Bonhoeffer-van der Pol model. This equation has been solved by a new Monte Carlo algorithm. We demonstrate that the ensuing distribution functions represent only the global characteristics of the underlying force field: lines of zero slope which attract nearby trajectories prove to be the regions of phase space where the distributions concentrate their amplitude. Since there are two such lines the distributions are bimodal representing repeated fluctuations between two lines of zero slope. Even in cases where the deterministic Bonhoeffer-van der Pol model does not show limit cycle behaviour the stochastic system produces a limit cycle. This cycle can be identified with the firing of neural pulses.     

Ionic mechanisms underlying tonic and phasic firing behaviors in retinal ganglion cells: A model study

In the retina, the firing behaviors that ganglion cells exhibit when exposed to light stimuli are very important due to the significant roles they play in encoding the visual information. However, the detailed mechanisms, especially the intrinsic properties that generate and modulate these firing behaviors is not completely clear yet. In this study, 2 typical firing behaviors—i.e., tonic and phasic activities, which are widely observed in retinal ganglion cells (RGCs)—are investigated. A modified computational model was developed to explore the possible ionic mechanisms that underlie the generation of these 2 firing patterns. Computational results indicate that the generation of tonic and phasic activities may be attributed to the collective actions of 2 kinds of adaptation currents, i.e., an inactivating sodium current and a delayed-rectifier potassium current. The concentration of magnesium ions has crucial but differential effects in the modulation of tonic and phasic firings, when the model neuron is driven by N-methyl-D-aspartate (NMDA) -type synaptic input instead of constant current injections. The proposed model has robust features that account for the ionic mechanisms underlying the tonic and phasic firing behaviors, and it may also be used as a good candidate for modeling some other firing patterns in RGCs.     

Development and modulation of endogenous bursting in identified neuron R15 of juvenile Aplysia

Evidence from a variety of both vertebrate and invertebrate preparations has demonstrated that modulation of the intrinsic firing patterns of individual neurons can have a dramatic effect on the functional output of a neural circuit. Although the mechanisms underlying the production and modulation of intrinsic firing patterns have been extensively studied in adult nervous systems, relatively little is known about how these two features of intrinsically active neurons develop. To address these issues, we have examined the development of endogenous bursting and its modulation by neuropeptides in the identified cell R15 of juvenile Aplysia. Confirming Ohmori (1981), we found that the mature parabolic bursting pattern of R15 is absent in early juvenile stages and develops only gradually over the last stage of juvenile development. We have then analyzed the modulatory effects of extracts made from the neurosecretory bag cells of Aplysia on the immature firing pattern of juvenile R15 cells. In the adult, neuroactive peptides released from the bag cells are known to intensify bursting. In juveniles, we have found that bag cell extract (BCE) can induce bursting prematurely as well as intensify immature bursts, whereas control extracts have no effect on the firing pattern of R15. These results show that the ionic currents necessary for the generation of endogenous bursting in R15 are present and can be modulated before the normal developmental expression of the burst pattern.     

Ionic mechanisms underlying tonic and phasic firing behaviors in retinal ganglion cells

In the retina, the firing behaviors that ganglion cells exhibit when exposed to light stimuli are very important due to the significant roles they play in encoding the visual information. However, the detailed mechanisms, especially the intrinsic properties that generate and modulate these firing behaviors is not completely clear yet. In this study, 2 typical firing behaviors—i.e., tonic and phasic activities, which are widely observed in retinal ganglion cells (RGCs)—are investigated. A modified computational model was developed to explore the possible ionic mechanisms that underlie the generation of these 2 firing patterns. Computational results indicate that the generation of tonic and phasic activities may be attributed to the collective actions of 2 kinds of adaptation currents, i.e., an inactivating sodium current and a delayed-rectifier potassium current. The concentration of magnesium ions has crucial but differential effects in the modulation of tonic and phasic firings, when the model neuron is driven by N-methyl-D-aspartate (NMDA) -type synaptic input instead of constant current injections. The proposed model has robust features that account for the ionic mechanisms underlying the tonic and phasic firing behaviors, and it may also be used as a good candidate for modeling some other firing patterns in RGCs.     

Modeling boundary vector cell firing given optic flow as a cue

Boundary vector cells in entorhinal cortex fire when a rat is in locations at a specific distance from walls of an environment. This firing may originate from memory of the barrier location combined with path integration, or the firing may depend upon the apparent visual input image stream. The modeling work presented here investigates the role of optic flow, the apparent change of patterns of light on the retina, as input for boundary vector cell firing. Analytical spherical flow is used by a template model to segment walls from the ground, to estimate self-motion and the distance and allocentric direction of walls, and to detect drop-offs. Distance estimates of walls in an empty circular or rectangular box have a mean error of less than or equal to two centimeters. Integrating these estimates into a visually driven boundary vector cell model leads to the firing patterns characteristic for boundary vector cells. This suggests that optic flow can influence the firing of boundary vector cells.     

Control of firing patterns by two transient potassium currents: leading spike,latency, bistability

Transient potassium currents distinctively affect firing properties, particularly in regulating the latency before repetitive firing. Pyramidal cells of the dorsal cochlear nucleus (DCN) have two transient potassium currents, I _Kif and I _Kis, fast and slowly inactivating, respectively, and they exhibit firing patterns with dramatically variable latencies. They show immediate repetitive firing, or only after a long latency with or without a leading spike, the so-called pauser and buildup patterns. We consider a conductance-based, ten-variable, single-compartment model for the DCN pyramidal cells (Kanold and Manis 2001). We develop and analyze a reduced three-variable integrate-and-fire model (KM-LIF) which captures the qualitative firing features. We apply dynamical systems methods to explain the underlying biophysical and mathematical mechanisms for the firing behaviors, including the characteristic firing patterns, the latency phase, the onset of repetitive firing, and some discontinuities in the timing of latency duration (e.i. first spike latency and first inter spike interval). Moreover, we obtain new insights associated with the leading spike by phase plane analysis. We further demonstrate the effects of possible heterogeneity of I _Kis. The latency before repetitive firing can be controlled to cover a large range by tuning of the relative amounts of I _Kif and I _Kis. Finally, we find for the full system robust bistability when enough I _Kis is present.     

A novel somatosensory spatial navigation system outside the hippocampal formation

Spatially selective firing of place cells, grid cells, boundary vector/border cells and head direction cells constitutes the basic building blocks of a canonical spatial navigation system centered on the hippocampal-entorhinal complex. While head direction cells can be found throughout the brain, spatial tuning outside the hippocampal formation is often non-specific or conjunctive to other representations such as a reward. Although the precise mechanism of spatially selective firing activity is not understood, various studies show sensory inputs, particularly vision, heavily modulate spatial representation in the hippocampal-entorhinal circuit. To better understand the contribution of other sensory inputs in shaping spatial representation in the brain, we performed recording from the primary somatosensory cortex in foraging rats. To our surprise, we were able to detect the full complement of spatially selective firing patterns similar to that reported in the hippocampal-entorhinal network, namely, place cells, head direction cells, boundary vector/border cells, grid cells and conjunctive cells, in the somatosensory cortex. These newly identified somatosensory spatial cells form a spatial map outside the hippocampal formation and support the hypothesis that location information modulates body representation in the somatosensory cortex. Our findings provide transformative insights into our understanding of how spatial information is processed and integrated in the brain, as well as functional operations of the somatosensory cortex in the context of rehabilitation with brain-machine interfaces.Subject terms: Biological techniques, Cell biology     

Modeling neural activity using the generalized inverse Gaussian distribution

Spike trains from neurons are often used to make inferences about the underlying processes that generate the spikes. Random walks or diffusions are commonly used to model these processes; in such models, a spike corresponds to the first passage of the diffusion to a boundary, or firing threshold. An important first step in such a study is to fit families of densities to the trains' interspike interval histograms; the estimated parameters, and the families' goodness of fit can then provide information about the process leading to the spikes. In this paper, we propose the generalized inverse Gaussian family because its members arise as first passage time distributions of certain diffusions to a constant boundary. We provide some theoretical support for the use of these diffusions in neural firing models. We compare this family with the lognormal family, using spike trains from retinal ganglion cells of goldfish, and simulations from an integrate-and-fire and a dynamical model for generating spikes. We show that the generalized inverse Gaussian family is closer to the true model in all these cases. Received: 16 September 1996 / Accepted in revised form: 2 July 1997     

Hippocampal Remapping Is Constrained by Sparseness rather than Capacity

Grid cells in the medial entorhinal cortex encode space with firing fields that are arranged on the nodes of spatial hexagonal lattices. Potential candidates to read out the space information of this grid code and to combine it with other sensory cues are hippocampal place cells. In this paper, we investigate a population of grid cells providing feed-forward input to place cells. The capacity of the underlying synaptic transformation is determined by both spatial acuity and the number of different spatial environments that can be represented. The codes for different environments arise from phase shifts of the periodical entorhinal cortex patterns that induce a global remapping of hippocampal place fields, i.e., a new random assignment of place fields for each environment. If only a single environment is encoded, the grid code can be read out at high acuity with only few place cells. A surplus in place cells can be used to store a space code for more environments via remapping. The number of stored environments can be increased even more efficiently by stronger recurrent inhibition and by partitioning the place cell population such that learning affects only a small fraction of them in each environment. We find that the spatial decoding acuity is much more resilient to multiple remappings than the sparseness of the place code. Since the hippocampal place code is sparse, we thus conclude that the projection from grid cells to the place cells is not using its full capacity to transfer space information. Both populations may encode different aspects of space.     

Models of place and grid cell firing and theta rhythmicity

Neuronal firing in the hippocampal formation (HF) of freely moving rodents shows striking examples of spatialorganization in the form of place, directional, boundary vector and grid cells. The firing of place and grid cells shows an intriguing form of temporal organization known as 'theta phase precession'. We review the mechanisms underlying theta phase precession of place cell firing, ranging from membrane potential oscillations to recurrent connectivity, and the relevant intra-cellular and extra-cellular data. We then consider the use of these models to explain the spatial structure of grid cell firing, and review the relevant intra-cellular and extra-cellular data. Finally, we consider the likely interaction between place cells, grid cells and boundary vector cells in estimating self-location as a compromise between path-integration and environmental information.     

Signal propagation in 2-dimensional threshold cellular space

Summary A two-dimensional network of uniformly connected McCulloch-Pitts neurons is considered and signal propagations in the network are analyzed. The problems are set up in the framework of cellular space such that each cell is a copy of any given McCulloch-Pitts neuron and is connected to the nearest neighboring cells. It is assumed that the threshold value is positive and that there exists only one firing cell at the beginning. Then it is shown that essentially there are only four signal propagation patterns and a firing pattern at any time t can be obtained by such a superposition of the propagation patterns that includes the newly defined concept of dominance and assimilation. The exact formulae representing firing patterns at any time t are obtained for any finite rectangle cell space with constant 0 (i.e., non-firing) boundary condition and for the entire two-dimensional cellular space.     

Consistent dynamics suggests tight regulation of biophysical parameters in a small network of bursting neurons

The neuronal firing patterns in the pyloric network of crustaceans are remarkably consistent among animals. Although this characteristic of the pyloric network is well-known, the biophysical mechanisms underlying the regulation of the systems output are receiving renewed attention. Computer simulations of the pyloric network recently demonstrated that consistent motor output can be achieved from neurons with disparate biophysical parameters among animals. Here we address this hypothesis by pharmacologically manipulating the pyloric network and analyzing the emerging voltage oscillations and firing patterns. Our results show that the pyloric network of the lobster stomatogastric ganglion maintains consistent and regular firing patterns even when entire populations of specific voltage-gated channels and synaptic receptors are blocked. The variations of temporal parameters used to characterize the burst patterns of the neurons as well as their intraburst spike dynamics do not display statistically significant increase after blocking the transient K-currents (with 4-aminopyridine), the glutamatergic inhibitory synapses (with picrotoxin), or the cholinergic synapses (with atropine) in pyloric networks from different animals. These data suggest that in this very compact circuit, the biophysical parameters are cell-specific and tightly regulated.     

A New Approach for Determining Phase Response Curves Reveals that Purkinje Cells Can Act as Perfect Integrators

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.     

The temporal program of chromosome replication: genomewide replication in clb5{Delta} Saccharomyces cerevisiae

Temporal regulation of origin activation is widely thought to explain the pattern of early- and late-replicating domains in the Saccharomyces cerevisiae genome. Recently, single-molecule analysis of replication suggested that stochastic processes acting on origins with different probabilities of activation could generate the observed kinetics of replication without requiring an underlying temporal order. To distinguish between these possibilities, we examined a clb5Delta strain, where origin firing is largely limited to the first half of S phase, to ask whether all origins nonspecifically show decreased firing (as expected for disordered firing) or if only some origins ("late" origins) are affected. Approximately half the origins in the mutant genome show delayed replication while the remainder replicate largely on time. The delayed regions can encompass hundreds of kilobases and generally correspond to regions that replicate late in wild-type cells. Kinetic analysis of replication in wild-type cells reveals broad windows of origin firing for both early and late origins. Our results are consistent with a temporal model in which origins can show some heterogeneity in both time and probability of origin firing, but clustering of temporally like origins nevertheless yields a genome that is organized into blocks showing different replication times.     

Optogenetic Modulation and Multi-Electrode Analysis of Cerebellar Networks In Vivo

The firing patterns of cerebellar Purkinje cells (PCs), as the sole output of the cerebellar cortex, determine and tune motor behavior. PC firing is modulated by various inputs from different brain regions and by cell-types including granule cells (GCs), climbing fibers and inhibitory interneurons. To understand how signal integration in PCs occurs and how subtle changes in the modulation of PC firing lead to adjustment of motor behaviors, it is important to precisely record PC firing in vivo and to control modulatory pathways in a spatio-temporal manner. Combining optogenetic and multi-electrode approaches, we established a new method to integrate light-guides into a multi-electrode system. With this method we are able to variably position the light-guide in defined regions relative to the recording electrode with micrometer precision. We show that PC firing can be precisely monitored and modulated by light-activation of channelrhodopsin-2 (ChR2) expressed in PCs, GCs and interneurons. Thus, this method is ideally suited to investigate the spatio/temporal modulation of PCs in anesthetized and in behaving mice.     

How Entorhinal Grid Cells May Learn Multiple Spatial Scales from a Dorsoventral Gradient of Cell Response Rates in a Self-organizing Map

Place cells in the hippocampus of higher mammals are critical for spatial navigation. Recent modeling clarifies how this may be achieved by how grid cells in the medial entorhinal cortex (MEC) input to place cells. Grid cells exhibit hexagonal grid firing patterns across space in multiple spatial scales along the MEC dorsoventral axis. Signals from grid cells of multiple scales combine adaptively to activate place cells that represent much larger spaces than grid cells. But how do grid cells learn to fire at multiple positions that form a hexagonal grid, and with spatial scales that increase along the dorsoventral axis? In vitro recordings of medial entorhinal layer II stellate cells have revealed subthreshold membrane potential oscillations (MPOs) whose temporal periods, and time constants of excitatory postsynaptic potentials (EPSPs), both increase along this axis. Slower (faster) subthreshold MPOs and slower (faster) EPSPs correlate with larger (smaller) grid spacings and field widths. A self-organizing map neural model explains how the anatomical gradient of grid spatial scales can be learned by cells that respond more slowly along the gradient to their inputs from stripe cells of multiple scales, which perform linear velocity path integration. The model cells also exhibit MPO frequencies that covary with their response rates. The gradient in intrinsic rhythmicity is thus not compelling evidence for oscillatory interference as a mechanism of grid cell firing. A response rate gradient combined with input stripe cells that have normalized receptive fields can reproduce all known spatial and temporal properties of grid cells along the MEC dorsoventral axis. This spatial gradient mechanism is homologous to a gradient mechanism for temporal learning in the lateral entorhinal cortex and its hippocampal projections. Spatial and temporal representations may hereby arise from homologous mechanisms, thereby embodying a mechanistic “neural relativity” that may clarify how episodic memories are learned.     

Blanes cells mediate persistent feedforward inhibition onto granule cells in the olfactory bulb

Inhibitory local circuits in the olfactory bulb play a critical role in determining the firing patterns of output neurons. However, little is known about the circuitry in the major plexiform layers of the olfactory bulb that regulate this output. Here we report the first electrophysiological recordings from Blanes cells, large stellate-shaped interneurons located in the granule cell layer. We find that Blanes cells are GABAergic and generate large I(CAN)-mediated afterdepolarizations following bursts of action potentials. Using paired two-photon guided intracellular recordings, we show that Blanes cells have a presumptive axon and monosynaptically inhibit granule cells. Sensory axon stimulation evokes barrages of EPSPs in Blanes cells that trigger long epochs of persistent spiking; this firing mode was reset by hyperpolarizing membrane potential steps. Persistent firing in Blanes cells may represent a novel mechanism for encoding short-term olfactory information through modulation of tonic inhibitory synaptic input onto bulbar neurons.     

Diversity of Intrinsic Frequency Encoding Patterns in Rat Cortical Neurons—Mechanisms and Possible Functions

Extracellular recordings of single neurons in primary and secondary somatosensory cortices of monkeys in vivo have shown that their firing rate can increase, decrease, or remain constant in different cells, as the external stimulus frequency increases. We observed similar intrinsic firing patterns (increasing, decreasing or constant) in rat somatosensory cortex in vitro, when stimulated with oscillatory input using conductance injection (dynamic clamp). The underlying mechanism of this observation is not obvious, and presents a challenge for mathematical modelling. We propose a simple principle for describing this phenomenon using a leaky integrate-and-fire model with sinusoidal input, an intrinsic oscillation and Poisson noise. Additional enhancement of the gain of encoding could be achieved by local network connections amongst diverse intrinsic response patterns. Our work sheds light on the possible cellular and network mechanisms underlying these opposing neuronal responses, which serve to enhance signal detection.     

Modulation of neuronal excitability by intracellular calcium buffering: from spiking to bursting

We have investigated the detailed regulation of neuronal firing pattern by the cytosolic calcium buffering capacity using a combination of mathematical modeling and patch-clamp recording in acute slice. Theoretical results show that a high calcium buffer concentration alters the characteristic regular firing of cerebellar granule cells and that a transition to various modes of oscillations occurs, including bursting. Using bifurcation analysis, we show that this transition from spiking to bursting is a consequence of the major slowdown of calcium dynamics. Patch-clamp recordings on cerebellar granule cells loaded with a high concentration of the fast calcium buffer BAPTA (15 mM) reveal dramatic alterations in their excitability as compared to cells loaded with 0.15 mM BAPTA. In high calcium buffering conditions, granule cells exhibit all bursting behaviors predicted by the model whereas bursting is never observed in low buffering. These results suggest that cytosolic calcium buffering capacity can tightly modulate neuronal firing patterns leading to generation of complex patterns and therefore that calcium-binding proteins may play a critical role in the non-synaptic plasticity and information processing in the central nervous system.