High Prevalence of Multistability of Rest States and Bursting in a Database of a Model Neuron

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance ()) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of . We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation.     

Theta oscillations by synaptic excitation in a neocortical circuit model

Neocortical theta-band oscillatory activity is associated with cognitive tasks involving learning and memory. This oscillatory activity is proposed to originate from the synchronization of interconnected layer V intrinsic bursting (IB) neurons by recurrent excitation. To test this hypothesis, a sparsely connected spiking circuit model based on empirical data was simulated using Hodgkin-Huxley-type bursting neurons and use-dependent depressing synaptic connections. In response to a heterogeneous tonic current stimulus, the model generated coherent and robust oscillatory activity throughout the theta-band (4-12 Hz). These oscillations were not, however, self-sustaining without a driving current, and not dependent on N-methyl-D-aspartate receptor synaptic currents. At realistic connection strengths, synaptic depression was necessary to avoid instability and expanded the basin of attraction for theta oscillations by controlling the gain of recurrent excitation. These results support the hypothesis that IB neuron networks can generate robust and coherent theta-band oscillations in neocortex.     

Oscillations and oscillatory behavior in small neural circuits

In order to determine the dynamical properties of central pattern generators (CPGs), we have examined the lobster stomatogastric ganglion using the tools of nonlinear dynamics. The lobster pyloric and gastric mill central pattern generators can be analyzed at both the cellular and network levels because they are small, i.e., contain only 25 neurons between them and each neuron and synapse are repeatedly identifiable from animal to animal. We discuss how the biophysical properties of each neuron and synapse in the two circuits act cooperatively to generate two different patterns of sequential activity, how these patterns are altered by neuromodulators and perturbed by noise and sensory inputs. Finally, we show how simplified Hindmarsh–Rose models can be made into analog electronic neurons that mimic the lobster neurons and in addition be incorporated into artificial CPGs with robotic applications.     

Six types of multistability in a neuronal model based on slow calcium current

Background

Multistability of oscillatory and silent regimes is a ubiquitous phenomenon exhibited by excitable systems such as neurons and cardiac cells. Multistability can play functional roles in short-term memory and maintaining posture. It seems to pose an evolutionary advantage for neurons which are part of multifunctional Central Pattern Generators to possess multistability. The mechanisms supporting multistability of bursting regimes are not well understood or classified.

Methodology/Principal Findings

Our study is focused on determining the bio-physical mechanisms underlying different types of co-existence of the oscillatory and silent regimes observed in a neuronal model. We develop a low-dimensional model typifying the dynamics of a single leech heart interneuron. We carry out a bifurcation analysis of the model and show that it possesses six different types of multistability of dynamical regimes. These types are the co-existence of 1) bursting and silence, 2) tonic spiking and silence, 3) tonic spiking and subthreshold oscillations, 4) bursting and subthreshold oscillations, 5) bursting, subthreshold oscillations and silence, and 6) bursting and tonic spiking. These first five types of multistability occur due to the presence of a separating regime that is either a saddle periodic orbit or a saddle equilibrium. We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.

Conclusions

We developed a neuronal model which exhibits a rich variety of different types of multistability. We described a novel mechanism supporting the bistability of bursting and silence. This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.     

Analog VLSI implementation of resonate-and-fire neuron

We propose an analog integrated circuit that implements a resonate-and-fire neuron (RFN) model based on the Lotka-Volterra (LV) system. The RFN model is a spiking neuron model that has second-order membrane dynamics, and thus exhibits fast damped subthreshold oscillation, resulting in the coincidence detection, frequency preference, and post-inhibitory rebound. The RFN circuit has been derived from the LV system to mimic such dynamical behavior of the RFN model. Through circuit simulations, we demonstrate that the RFN circuit can act as a coincidence detector and a band-pass filter at circuit level even in the presence of additive white noise and background random activity. These results show that our circuit is expected to be useful for very large-scale integration (VLSI) implementation of functional spiking neural networks.     

Responses of a bursting pacemaker to excitation reveal spatial segregation between bursting and spiking mechanisms

Central pattern generators (CPGs) frequently include bursting neurons that serve as pacemakers for rhythm generation. Phase resetting curves (PRCs) can provide insight into mechanisms underlying phase locking in such circuits. PRCs were constructed for a pacemaker bursting complex in the pyloric circuit in the stomatogastric ganglion of the lobster and crab. This complex is comprised of the Anterior Burster (AB) neuron and two Pyloric Dilator (PD) neurons that are all electrically coupled. Artificial excitatory synaptic conductance pulses of different strengths and durations were injected into one of the AB or PD somata using the Dynamic Clamp. Previously, we characterized the inhibitory PRCs by assuming a single slow process that enabled synaptic inputs to trigger switches between an up state in which spiking occurs and a down state in which it does not. Excitation produced five different PRC shapes, which could not be explained with such a simple model. A separate dendritic compartment was required to separate the mechanism that generates the up and down phases of the bursting envelope (1) from synaptic inputs applied at the soma, (2) from axonal spike generation and (3) from a slow process with a slower time scale than burst generation. This study reveals that due to the nonlinear properties and compartmentalization of ionic channels, the response to excitation is more complex than inhibition.     

Dissecting cascade computational components in spiking neural networks

Finding out the physical structure of neuronal circuits that governs neuronal responses is an important goal for brain research. With fast advances for large-scale recording techniques, identification of a neuronal circuit with multiple neurons and stages or layers becomes possible and highly demanding. Although methods for mapping the connection structure of circuits have been greatly developed in recent years, they are mostly limited to simple scenarios of a few neurons in a pairwise fashion; and dissecting dynamical circuits, particularly mapping out a complete functional circuit that converges to a single neuron, is still a challenging question. Here, we show that a recent method, termed spike-triggered non-negative matrix factorization (STNMF), can address these issues. By simulating different scenarios of spiking neural networks with various connections between neurons and stages, we demonstrate that STNMF is a persuasive method to dissect functional connections within a circuit. Using spiking activities recorded at neurons of the output layer, STNMF can obtain a complete circuit consisting of all cascade computational components of presynaptic neurons, as well as their spiking activities. For simulated simple and complex cells of the primary visual cortex, STNMF allows us to dissect the pathway of visual computation. Taken together, these results suggest that STNMF could provide a useful approach for investigating neuronal systems leveraging recorded functional neuronal activity.     

Running as fast as it can: How spiking dynamics form object groupings in the laminar circuits of visual cortex

How spiking neurons cooperate to control behavioral processes is a fundamental problem in computational neuroscience. Such cooperative dynamics are required during visual perception when spatially distributed image fragments are grouped into emergent boundary contours. Perceptual grouping is a challenge for spiking cells because its properties of collinear facilitation and analog sensitivity occur in response to binary spikes with irregular timing across many interacting cells. Some models have demonstrated spiking dynamics in recurrent laminar neocortical circuits, but not how perceptual grouping occurs. Other models have analyzed the fast speed of certain percepts in terms of a single feedforward sweep of activity, but cannot explain other percepts, such as illusory contours, wherein perceptual ambiguity can take hundreds of milliseconds to resolve by integrating multiple spikes over time. The current model reconciles fast feedforward with slower feedback processing, and binary spikes with analog network-level properties, in a laminar cortical network of spiking cells whose emergent properties quantitatively simulate parametric data from neurophysiological experiments, including the formation of illusory contours; the structure of non-classical visual receptive fields; and self-synchronizing gamma oscillations. These laminar dynamics shed new light on how the brain resolves local informational ambiguities through the use of properly designed nonlinear feedback spiking networks which run as fast as they can, given the amount of uncertainty in the data that they process.     

Coexistence of Tonic Spiking Oscillations in a Leech Neuron Model

The leech neuron model studied here has a remarkable dynamical plasticity. It exhibits a wide range of activities including various types of tonic spiking and bursting. In this study we apply methods of the qualitative theory of dynamical systems and the bifurcation theory to analyze the dynamics of the leech neuron model with emphasis on tonic spiking regimes. We show that the model can demonstrate bi-stability, such that two modes of tonic spiking coexist. Under a certain parameter regime, both tonic spiking modes are represented by the periodic attractors. As a bifurcation parameter is varied, one of the attractors becomes chaotic through a cascade of period-doubling bifurcations, while the other remains periodic. Thus, the system can demonstrate co-existence of a periodic tonic spiking with either periodic or chaotic tonic spiking. Pontryagins averaging technique is used to locate the periodic orbits in the phase space.     

Synchronous and asynchronous bursting states: role of intrinsic neural dynamics

Brain signals such as local field potentials often display gamma-band oscillations (30-70 Hz) in a variety of cognitive tasks. These oscillatory activities possibly reflect synchronization of cell assemblies that are engaged in a cognitive function. A type of pyramidal neurons, i.e., chattering neurons, show fast rhythmic bursting (FRB) in the gamma frequency range, and may play an active role in generating the gamma-band oscillations in the cerebral cortex. Our previous phase response analyses have revealed that the synchronization between the coupled bursting neurons significantly depends on the bursting mode that is defined as the number of spikes in each burst. Namely, a network of neurons bursting through a Ca(2+)-dependent mechanism exhibited sharp transitions between synchronous and asynchronous firing states when the neurons exchanged the bursting mode between singlet, doublet and so on. However, whether a broad class of bursting neuron models commonly show such a network behavior remains unclear. Here, we analyze the mechanism underlying this network behavior using a mathematically tractable neuron model. Then we extend our results to a multi-compartment version of the NaP current-based neuron model and prove a similar tight relationship between the bursting mode changes and the network state changes in this model. Thus, the synchronization behavior couples tightly to the bursting mode in a wide class of networks of bursting neurons.     

Improving noise resistance of intrinsic rhythms in a square-wave burster model

The square-wave burster (Wang and Rinzel, 2003) is a class of autonomous bursting cells that share a bifurcation structure. It is known that this class of cells is involved in the generation of various life-supporting rhythms. In our research to realize an electronic circuit that mimics the rhythm generating mechanism in the square-wave burster, our circuit experimentally exhibited severe fluctuations in its rhythmic activity. We have found a noise-sensitive region in the phase portrait of the ideal model and have proposed modifications of the model that can reduce this fluctuation. A possible modification to ionic-conductance neuron models (Kohno and Aihara, 2011) was inspired by them. This modification, however, cannot be applied to a group of square-wave bursters, including the Butera–Rinzel–Smith model (0010 and 0050), which is a model of the pre-Bötzinger complex bursting neuron that plays a crucial role in the generation of respiration rhythms, because this modification premises that the slow dynamics originates from an activation gate variable of a hyperpolarizing ionic current. However, in some square-wave bursters, they are controlled by an inactivation gate variable of a depolarizing ionic current. In this study, we proposed a similar modification with a completely different mechanism that can be applied to this group of square-wave bursters. In the presence of noises, the modified Butera–Rinzel–Smith model can generate rhythmic activity that is more stable and similar to biological observations than the original model. The mechanisms underlying this modification are explained with noisy bifurcation diagrams.     

State-Dependent Effects of Na Channel Noise on Neuronal Burst Generation

We explore the effects of stochastic sodium (Na) channel activation on the variability and dynamics of spiking and bursting in a model neuron. The complete model segregates Hodgin-Huxley-type currents into two compartments, and undergoes applied current-dependent bifurcations between regimes of periodic bursting, chaotic bursting, and tonic spiking. Noise is added to simulate variable, finite sizes of the population of Na channels in the fast spiking compartment.During tonic firing, Na channel noise causes variability in interspike intervals (ISIs). The variance, as well as the sensitivity to noise, depend on the model's biophysical complexity. They are smallest in an isolated spiking compartment; increase significantly upon coupling to a passive compartment; and increase again when the second compartment also includes slow-acting currents. In this full model, sufficient noise can convert tonic firing into bursting.During bursting, the actions of Na channel noise are state-dependent. The higher the noise level, the greater the jitter in spike timing within bursts. The noise makes the burst durations of periodic regimes variable, while decreasing burst length duration and variance in a chaotic regime. Na channel noise blurs the sharp transitions of spike time and burst length seen at the bifurcations of the noise-free model. Close to such a bifurcation, the burst behaviors of previously periodic and chaotic regimes become essentially indistinguishable.We discuss biophysical mechanisms, dynamical interpretations and physiological implications. We suggest that noise associated with finite populations of Na channels could evoke very different effects on the intrinsic variability of spiking and bursting discharges, depending on a biological neuron's complexity and applied current-dependent state. We find that simulated channel noise in the model neuron qualitatively replicates the observed variability in burst length and interburst interval in an isolated biological bursting neuron.     

Firing patterns in the adaptive exponential integrate-and-fire model

For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.     

Mechanisms of firing patterns in fast-spiking cortical interneurons

Cortical fast-spiking (FS) interneurons display highly variable electrophysiological properties. Their spike responses to step currents occur almost immediately following the step onset or after a substantial delay, during which subthreshold oscillations are frequently observed. Their firing patterns include high-frequency tonic firing and rhythmic or irregular bursting (stuttering). What is the origin of this variability? In the present paper, we hypothesize that it emerges naturally if one assumes a continuous distribution of properties in a small set of active channels. To test this hypothesis, we construct a minimal, single-compartment conductance-based model of FS cells that includes transient Na(+), delayed-rectifier K(+), and slowly inactivating d-type K(+) conductances. The model is analyzed using nonlinear dynamical system theory. For small Na(+) window current, the neuron exhibits high-frequency tonic firing. At current threshold, the spike response is almost instantaneous for small d-current conductance, gd, and it is delayed for larger gd. As gd further increases, the neuron stutters. Noise substantially reduces the delay duration and induces subthreshold oscillations. In contrast, when the Na(+) window current is large, the neuron always fires tonically. Near threshold, the firing rates are low, and the delay to firing is only weakly sensitive to noise; subthreshold oscillations are not observed. We propose that the variability in the response of cortical FS neurons is a consequence of heterogeneities in their gd and in the strength of their Na(+) window current. We predict the existence of two types of firing patterns in FS neurons, differing in the sensitivity of the delay duration to noise, in the minimal firing rate of the tonic discharge, and in the existence of subthreshold oscillations. We report experimental results from intracellular recordings supporting this prediction.     

Generating oscillatory bursts from a network of regular spiking neurons without inhibition

Avian nucleus isthmi pars parvocellularis (Ipc) neurons are reciprocally connected with the layer 10 (L10) neurons in the optic tectum and respond with oscillatory bursts to visual stimulation. Our in vitro experiments show that both neuron types respond with regular spiking to somatic current injection and that the feedforward and feedback synaptic connections are excitatory, but of different strength and time course. To elucidate mechanisms of oscillatory bursting in this network of regularly spiking neurons, we investigated an experimentally constrained model of coupled leaky integrate-and-fire neurons with spike-rate adaptation. The model reproduces the observed Ipc oscillatory bursting in response to simulated visual stimulation. A scan through the model parameter volume reveals that Ipc oscillatory burst generation can be caused by strong and brief feedforward synaptic conductance changes. The mechanism is sensitive to the parameter values of spike-rate adaptation. In conclusion, we show that a network of regular-spiking neurons with feedforward excitation and spike-rate adaptation can generate oscillatory bursting in response to a constant input.     

Analysis of an autonomous phase model for neuronal parabolic bursting

An understanding of the nonlinear dynamics of bursting is fundamental in unraveling structure-function relations in nerve and secretory tissue. Bursting is characterized by alternations between phases of rapid spiking and slowly varying potential. A simple phase model is developed to study endogenous parabolic bursting, a class of burst activity observed experimentally in excitable membrane. The phase model is motivated by Rinzel and Lee's dissection of a model for neuronal parabolic bursting (J. Math. Biol. 25, 653–675 (1987)). Rapid spiking is represented canonically by a one-variable phase equation that is coupled bi-directionally to a two-variable slow system. The model is analyzed in the slow-variable phase plane, using quasi steady-state assumptions and formal averaging. We derive a reduced system to explore where the full model exhibits bursting, steady-states, continuous and modulated spiking. The relative speed of activation and inactivation of the slow variables strongly influences the burst pattern as well as other dynamics. We find conditions of the bistability of solutions between continuous spiking and bursting. Although the phase model is simple, we demonstrate that it captures many dynamical features of more complex biophysical models.This research was partially supported by NSF-JOINT RESEARCH grant 8803573, grant from CONCYT and DGAPA(UNAM) Mexico for H. Carrillo, and for the S. M. Baer NSF DMS-9107538     

Travelling wave patterns in a model of the spinal pattern generator using spiking neurons

The aim of this study is to produce travelling waves in a planar net of artificial spiking neurons. Provided that the parameters of the waves – frequency, wavelength and orientation – can be sufficiently controlled, such a network can serve as a model of the spinal pattern generator for swimming and terrestrial quadruped locomotion. A previous implementation using non-spiking, sigmoid neurons lacked the physiological plausibility that can only be attained using more realistic spiking neurons. Simulations were conducted using three types of spiking neuronal models. First, leaky integrate-and-fire neurons were used. Second, we introduced a phenomenological bursting neuron. And third, a canonical model neuron was implemented which could reproduce the full dynamics of the Hodgkin–Huxley neuron. The conditions necessary to produce appropriate travelling waves corresponded largely to the known anatomy and physiology of the spinal cord. Especially important features for the generation of travelling waves were the topology of the local connections – so-called off-centre connectivity – the availability of dynamic synapses and, to some extent, the availability of bursting cell types. The latter were necessary to produce stable waves at the low frequencies observed in quadruped locomotion. In general, the phenomenon of travelling waves was very robust and largely independent of the network parameters and emulated cell types.     

Cell Type-Specific Separation of Subicular Principal Neurons during Network Activities

The hippocampal output structure, the subiculum, expresses two major memory relevant network rhythms, sharp wave ripple and gamma frequency oscillations. To this date, it remains unclear how the two distinct types of subicular principal cells, intrinsically bursting and regular spiking neurons, participate in these two network rhythms. Using concomitant local field potential and intracellular recordings in an in vitro mouse model that allows the investigation of both network rhythms, we found a cell type-specific segregation of principal neurons into participating intrinsically bursting and non-participating regular spiking cells. However, if regular spiking cells were kept at a more depolarized level, they did participate in a specific manner, suggesting a potential bimodal working model dependent on the level of excitation. Furthermore, intrinsically bursting and regular spiking cells exhibited divergent intrinsic membrane and synaptic properties in the active network. Thus, our results suggest a cell-type-specific segregation of principal cells into two separate groups during network activities, supporting the idea of two parallel streams of information processing within the subiculum.     

Spike timing precision of neuronal circuits

Spike timing is believed to be a key factor in sensory information encoding and computations performed by the neurons and neuronal circuits. However, the considerable noise and variability, arising from the inherently stochastic mechanisms that exist in the neurons and the synapses, degrade spike timing precision. Computational modeling can help decipher the mechanisms utilized by the neuronal circuits in order to regulate timing precision. In this paper, we utilize semi-analytical techniques, which were adapted from previously developed methods for electronic circuits, for the stochastic characterization of neuronal circuits. These techniques, which are orders of magnitude faster than traditional Monte Carlo type simulations, can be used to directly compute the spike timing jitter variance, power spectral densities, correlation functions, and other stochastic characterizations of neuronal circuit operation. We consider three distinct neuronal circuit motifs: Feedback inhibition, synaptic integration, and synaptic coupling. First, we show that both the spike timing precision and the energy efficiency of a spiking neuron are improved with feedback inhibition. We unveil the underlying mechanism through which this is achieved. Then, we demonstrate that a neuron can improve on the timing precision of its synaptic inputs, coming from multiple sources, via synaptic integration: The phase of the output spikes of the integrator neuron has the same variance as that of the sample average of the phases of its inputs. Finally, we reveal that weak synaptic coupling among neurons, in a fully connected network, enables them to behave like a single neuron with a larger membrane area, resulting in an improvement in the timing precision through cooperation.