Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes

 The Hodgkin–Huxley equations with a slight modification are investigated, in which the inactivation process (h) of sodium channels or the activation process of potassium channels (n) is slowed down. We show that the equations produce a variety of action potential waveforms ranging from a plateau potential, such as in heart muscle cells, to chaotic bursting firings. When h is slowed down – differently from the case of n variable being slow – chaotic bursting oscillations are observed for a wide range of parameter values although both variables cause a decrease in the membrane potential. The underlying nonlinear dynamics of various action potentials are analyzed using bifurcation theory and a so-called slow–fast decomposition analysis. It is shown that a simple topological property of the equilibrium curves of slow and fast subsystems is essential to the production of chaotic oscillations, and this is the cause of the large difference in global firing characteristics between the h-slow and n-slow cases. Received: 9 August 2000 / Accepted in revised form: 10 January 2001     

Waves, bumps, and patterns in neural field theories

Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.     

Firing pattern of bursting neurons under sinusoidal drive in mean-field modeling

Bursting has been observed in many sensory neurons, and is thought to be important in neural signaling, sleep, and some disorders of the brain. Bursting neurons have been studied via various types of conductance-based models at the single-neuron level. Important features of bursting have been reproduced by this type of model, but it is not certain how well the behavior of populations of bursting neurons can be represented solely by that of individual neurons. To study bursting neurons at the population level, a conductance-based model is incorporated into a mean-field model to yield a mean-field bursting model. The responses of the model to sinusoidal inputs are studied, showing that neurons with various different initial states are capable of phase-locked or intermittent firing, depending on their baseline voltage. Furthermore, depending on this voltage, the bursting frequency either slaves to the original unperturbed bursting frequency or approaches a steady value when the external driving frequency increases. Finally, use of white noise perturbations shows that the bursting frequency of the neurons remains the same even under a more general external stimulus.     

Ion concentration dynamics as a mechanism for neuronal bursting

We describe a simple conductance-based model neuron that includes intra- and extracellular ion concentration dynamics and show that this model exhibits periodic bursting. The bursting arises as the fast-spiking behavior of the neuron is modulated by the slow oscillatory behavior in the ion concentration variables and vice versa. By separating these time scales and studying the bifurcation structure of the neuron, we catalog several qualitatively different bursting profiles that are strikingly similar to those seen in experimental preparations. Our work suggests that ion concentration dynamics may play an important role in modulating neuronal excitability in real biological systems.     

Control of computational dynamics of coupled integrate-and-fire neurons

 Generation and control of different dynamical modes of computational processes in a net of interconnected integrate-and-fire neurons are demonstrated. A net architecture resembling a generic cortical structure is formed from pairs of excitatory and inhibitory units with excitatory connections between and inhibitory connections within pairs. Integrate-and-fire model neurons derived from detailed conductance-based models of neocortical pyramidal cells and fast-spiking interneurons are employed for the excitatory and inhibitory units, respectively. Firing-rate adaptation is incorporated into the excitatory units based on the regulation of the slow afterhyperpolarization phase of action potentials by intracellular calcium ions. Saturation of synaptic conductances is implemented for the interconnections between units. It is shown that neuronal adaptation of the excitatory units can generate richer net dynamics than relaxation to fixed-point attractors in a pattern space. At strong adaptivity, i.e. when the neuronal excitability is strongly influenced by the preceding activity, complex dynamics of either aperiodic or limit-cycle character are generated in both the pattern space and the phase space of all dynamical variables. This regime corresponds to an exploratory mode of the system, in which the pattern space can be searched. At weak adaptivity, the dynamics are governed by fixed-point attractors in the pattern space, and this corresponds to a mode for retrieval of a particular pattern. In the brain, neuronal adaptivity can be regulated by various neuromodulators. The results are in accordance with those recently obtained by means of more abstract models formulated in terms of mean firing rates. The increased realism makes the present model reveal more detailed mechanisms and strengthens the relevance of the conclusions to biological systems. The simplicity and realism of the coupled integrate-and-fire neurons make the present model useful for studies of systems in which the temporal aspects of neural coding are important. Received: 8 December 1995 / Accepted in revised form: 23 January 1997     

A Hamilton-Jacobi-Bellman approach for termination of seizure-like bursting

We use Hamilton-Jacobi-Bellman methods to find minimum-time and energy-optimal control strategies to terminate seizure-like bursting behavior in a conductance-based neural model. Averaging is used to eliminate fast variables from the model, and a target set is defined through bifurcation analysis of the slow variables of the model. This method is illustrated for a single neuron model and for a network model to illustrate its efficacy in terminating bursting once it begins. This work represents a numerical proof-of-concept that a new class of control strategies can be employed to mitigate bursting, and could ultimately be adapted to treat medically intractible epilepsy in patient-specific models.     

Low dimensional model of bursting neurons

A computationally efficient, biophysically-based model of neuronal behavior is presented; it incorporates ion channel dynamics in its two fast ion channels while preserving simplicity by representing only one slow ion current. The model equations are shown to provide a wide array of physiological dynamics in terms of spiking patterns, bursting, subthreshold oscillations, and chaotic firing. Despite its simplicity, the model is capable of simulating an extensive range of spiking patterns. Several common neuronal behaviors observed in vivo are demonstrated by varying model parameters. These behaviors are classified into dynamical classes using phase diagrams whose boundaries in parameter space prove to be accurately delineated by linear stability analysis. This simple model is suitable for use in large scale simulations involving neural field theory or neuronal networks.     

Neural assemblies revealed by inferred connectivity-based models of prefrontal cortex recordings

We extend the theory of weakly coupled oscillators to incorporate slowly varying inputs and parameters. We employ a combination of regular perturbation and an adiabatic approximation to derive equations for the phase-difference between a pair of oscillators. We apply this to the simple Hopf oscillator and then to a biophysical model. The latter represents the behavior of a neuron that is subject to slow modulation of a muscarinic current such as would occur during transient attention through cholinergic activation. Our method extends and simplifies the recent work of Kurebayashi (Physical Review Letters, 111, 214101, 2013) to include coupling. We apply the method to an all-to-all network and show that there is a waxing and waning of synchrony of modulated neurons.     

Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map

 We discuss a method by which the dynamics of a network of neurons, coupled by mutual inhibition, can be reduced to a one-dimensional map. This network consists of a pair of neurons, one of which is an endogenous burster, and the other excitable but not bursting in the absence of phasic input. The latter cell has more than one slow process. The reduction uses the standard separation of slow/fast processes; it also uses information about how the dynamics on the slow manifold evolve after a finite amount of slow time. From this reduction we obtain a one-dimensional map dependent on the parameters of the original biophysical equations. In some parameter regimes, one can deduce that the original equations have solutions in which the active phase of the originally excitable cell is constant from burst to burst, while in other parameter regimes it is not. The existence or absence of this kind of regulation corresponds to qualitatively different dynamics in the one-dimensional map. The computations associated with the reduction and the analysis of the dynamics includes the use of coordinates that parameterize by time along trajectories, and “singular Poincaré maps” that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold. Received: 19 May 1997 / Revised version: 6 April 1998     

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.     

A model of the electrophysiological properties of nucleus reticularis thalami neurons.

A model of the electrophysiological properties of rodent nucleus reticularis thalami (NRT) neurons of the dorsal lateral thalamus was developed using Hodgkin-Huxley style equations. The model incorporated voltage-dependent rate constants and kinetics obtained from recent voltage-clamp experiments in vitro. The intrinsic electroresponsivity of the model cell was found to be similar to several empirical observations. Three distinct modes of oscillatory activity were identified: 1) a pattern of slow rhythmic burst firing (0.5-7 Hz) usually associated with membrane potentials negative to approximately -70 mV which resulted from the interplay of ITs and IK(Ca); 2) at membrane potentials from approximately -69 to -62 mV, rhythmic burst firing in the spindle frequency range (7-12 Hz) developed and was immediately followed by a tonic tail of single spike firing after several bursts. The initial bursting rhythm resulted from the interaction of ITs and IK(Ca), with a slow after-depolarization due to ICAN which mediated the later tonic firing; 3) with further depolarization of the membrane potential positive to approximately -61 mV, sustained tonic firing appeared in the 10-200-Hz frequency range depending on the amplitude of the injected current. The frequency of this firing was also dependent on the maximum conductance of the leak current, IK(leak), and an interaction between the fast currents involved in generating action potentials, INa(fast) and IK(DR), and the persistent Na+ current, INa(P). Transitions between different firing modes were identified and studied parametrically.     

Inactivation of HIT cell Ca2+ current by a simulated burst of Ca2+ action potentials.

A novel voltage-clamp protocol was developed to test whether slow inactivation of Ca2+ current occurs during bursting in insulin-secreting cells. Single insulin-secreting HIT cells were patch-clamped and their Ca2+ currents were isolated pharmacologically. A computed beta-cell burst was used as a voltage-clamp command and the net Ca2+ current elicited was determined as a cadmium difference current. Ca2+ current rapidly activated during the computed plateau and spike depolarizations and then slowly decayed. Integration of this Ca2+ current yielded an estimate of total Ca influx. To further analyze Ca2+ current inactivation during a burst, repetitive test pulses to + 10 mV were added to the voltage command. Current elicited by these pulses was constant during the interburst, but then slowly and reversibly decreased during the depolarizing plateau. This inactivation was reduced by replacing external Ca2+ with Ba2+ as a charge carrier, and in some cells inactivation was slower in Ba2+. Experimental results were compared with the predictions of the Keizer-Smolen mathematical model of bursting, after subjecting model equations to identical voltage commands. In this model, bursting is driven by the slow, voltage-dependent inactivation of Ca current during the plateau active phase. The K-S model could account for the slope of the slow decay of spike-elicited Ca current, the waveform of individual Ca current spikes, and the suppression of test pulse-elicited Ca current during a burst command. However, the extent and rate of fast inactivation were underestimated by the model.(ABSTRACT TRUNCATED AT 250 WORDS)     

A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells

The electrical activity of endocrine pituitary cells is mediated by a plethora of ionic currents and establishing the role of a single channel type is difficult. Experimental observations have shown however that fast-activating voltage- and calcium-dependent potassium (BK) current tends to promote bursting in pituitary cells. This burst promoting effect requires fast activation of the BK current, otherwise it is inhibitory to bursting. In this work, we analyze a pituitary cell model in order to answer the question of why the BK activation must be fast to promote bursting. We also examine how the interplay between the activation rate and conductance of the BK current shapes the bursting activity. We use the multiple timescale structure of the model to our advantage and employ geometric singular perturbation theory to demonstrate the origin of the bursting behaviour. In particular, we show that the bursting can arise from either canard dynamics or slow passage through a dynamic Hopf bifurcation. We then compare our theoretical predictions with experimental data using the dynamic clamp technique and find that the data is consistent with a burst mechanism due to a slow passage through a Hopf.     

The assembly of ionic currents in a thalamic neuron. I. The three-dimensional model

We have previously discussed qualitative models for bursting and thalamic neurons that were obtained by modifying a simple two-dimensional model for repetitive firing. In this paper we report the results of making a similar sequence of modifications to a more elaborate six-dimensional model of repetitive firing which is based on the Hodgkin-Huxley equations. To do this we first reduce the six-dimensional model to a two-dimensional model that resembles our original two-dimensional qualitative model. This is achieved by defining a new variable, which we call q. We then add a subthreshold inward current and a subthreshold outward current having a variable, z, that changes slowly. This gives a three-dimensional (v,q,z) model of the Hodgkin-Huxley type, which we refer to as the z-model. Depending on the choice of parameter values this model resembles our previous models of bursting and thalamic neurons. At each stage in the development of these models we return to the corresponding seven-dimensional model to confirm that we can obtain similar solutions by using the complete system of equations. The analysis of the three-dimensional model involves a state diagram and a stability diagram. The state diagram shows the projection of the phase path from v,q,z space into the v,z plane, together with the projections of the curves z = 0 and v = q = 0. The stability of the points on the curve v = q = 0, which we call the v, q nullcurve, is determined by the stability diagram. Taken together the state and stability diagrams show how to assemble the ionic currents to produce a given firing pattern.     

A comparative analysis of multi-conductance neuronal models <Emphasis Type="Italic">in silico</Emphasis>

We demonstrate that a previously presented flexible silicon–neuron architecture can implement three disparate conductance-based neuron models with both fast and slow dynamics. By exploiting the real-time nature of this physical implementation, we mapped the model dynamics across a large region of parameter space. We also found that two of these dynamically different models represent points in a contiguous bursting space that spans between the two models. By systematically varying the model parameters, we also found that multiple, diverse trajectories in parameter space connected the two canonical bursting points. In addition, we found that the combination of parameter values keeps the neuron in the bursting region. These findings demonstrate the usefulness of the silicon–neuron architecture as a neural-modeling tool and illustrate its versatility as a platform for a multi-behavioral neuron that resembles its living analog.     

Slow variable dominance and phase resetting in phantom bursting

Bursting oscillations are common in neurons and endocrine cells. One type of bursting model with two slow variables has been called ‘phantom bursting’ since the burst period is a blend of the time constants of the slow variables. A phantom bursting model can produce bursting with a wide range of periods: fast (short period), medium, and slow (long period). We describe a measure, which we call the ‘dominance factor’, of the relative contributions of the two slow variables to the bursting produced by a simple phantom bursting model. Using this tool, we demonstrate how the control of different phases of the burst can be shifted from one slow variable to another by changing a model parameter. We then show that the dominance curves obtained as a parameter is varied can be useful in making predictions about the resetting properties of the model cells. Finally, we demonstrate two mechanisms by which phase-independent resetting of a burst can be achieved, as has been shown to occur in the electrical activity of pancreatic islets.     

Classification of bursting patterns: A tale of two ducks

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product, it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram and colleagues, and then by Golubitsky and colleagues, which, together with the Rinzel-Izhikevich proposals, provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least 2 slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the 2 main families of folded-node bursters, depending upon the phase (active/spiking or silent/nonspiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast subsystem approach.     

Time-dependent changes in membrane excitability during glucose-induced bursting activity in pancreatic β cells

In our companion paper, the physiological functions of pancreatic β cells were analyzed with a new β-cell model by time-based integration of a set of differential equations that describe individual reaction steps or functional components based on experimental studies. In this study, we calculate steady-state solutions of these differential equations to obtain the limit cycles (LCs) as well as the equilibrium points (EPs) to make all of the time derivatives equal to zero. The sequential transitions from quiescence to burst-interburst oscillations and then to continuous firing with an increasing glucose concentration were defined objectively by the EPs or LCs for the whole set of equations. We also demonstrated that membrane excitability changed between the extremes of a single action potential mode and a stable firing mode during one cycle of bursting rhythm. Membrane excitability was determined by the EPs or LCs of the membrane subsystem, with the slow variables fixed at each time point. Details of the mode changes were expressed as functions of slowly changing variables, such as intracellular [ATP], [Ca(2+)], and [Na(+)]. In conclusion, using our model, we could suggest quantitatively the mutual interactions among multiple membrane and cytosolic factors occurring in pancreatic β cells.     

Resonances and noise in a stochastic Hindmarsh-Rose model of thalamic neurons

Thalamic neurons exhibit subthreshold resonance when stimulated with small sine wave signals of varying frequency and stochastic resonance when noise is added to these signals. We study a stochastic Hindmarsh-Rose model using Monte-Carlo simulations to investigate how noise, in conjunction with subthreshold resonance, leads to a preferred frequency in the firing pattern. The resulting stochastic resonance (SR) exhibits a preferred firing frequency that is approximately exponential in its dependence on the noise amplitude. In similar experiments, frequency dependent SR is found in the reliability of detection of alpha-function inputs under noise, which are more realistic inputs for neurons. A mathematical analysis of the equations reveals that the frequency preference arises from the dynamics of the slow variable. Noise can then transfer the resonance over the firing threshold because of the proximity of the fast subsystem to a Hopf bifurcation point. Our results may have implications for the behavior of thalamic neurons in a network, with noise switching the membrane potential between different resonance modes.