Frequency entrainment of squid axon membrane

When less is more: non-monotonic spike sequence processing in neurons

Fundamental response properties of neurons centrally underly the computational capabilities of both individual nerve cells and neural networks. Most studies on neuronal input-output relations have focused on continuous-time inputs such as constant or noisy sinusoidal currents. Yet, most neurons communicate via exchanging action potentials (spikes) at discrete times. Here, we systematically analyze the stationary spiking response to regular spiking inputs and reveal that it is generically non-monotonic. Our theoretical analysis shows that the underlying mechanism relies solely on a combination of the discrete nature of the communication by spikes, the capability of locking output to input spikes and limited resources required for spike processing. Numerical simulations of mathematically idealized and biophysically detailed models, as well as neurophysiological experiments confirm and illustrate our theoretical predictions.

Neurodynamics, tonality, and the auditory brainstem response

Tonal relationships are foundational in music, providing the basis upon which musical structures, such as melodies, are constructed and perceived. A recent dynamic theory of musical tonality predicts that networks of auditory neurons resonate nonlinearly to musical stimuli. Nonlinear resonance leads to stability and attraction relationships among neural frequencies, and these neural dynamics give rise to the perception of relationships among tones that we collectively refer to as tonal cognition. Because this model describes the dynamics of neural populations, it makes specific predictions about human auditory neurophysiology. Here, we show how predictions about the auditory brainstem response (ABR) are derived from the model. To illustrate, we derive a prediction about population responses to musical intervals that has been observed in the human brainstem. Our modeled ABR shows qualitative agreement with important features of the human ABR. This provides a source of evidence that fundamental principles of auditory neurodynamics might underlie the perception of tonal relationships, and forces reevaluation of the role of learning and enculturation in tonal cognition.

Bistability and resonance in the periodically stimulated Hodgkin-Huxley model with noise

We describe general characteristics of the Hodgkin-Huxley neuron's response to a periodic train of short current pulses with Gaussian noise. The deterministic neuron is bistable for antiresonant frequencies. When the stimuli arrive at the resonant frequency the firing rate is a continuous function of the current amplitude I(0) and scales as (I(0)-I(th))(1/2), characteristic of a saddle-node bifurcation at the threshold I(th). Intervals of continuous irregular response alternate with integer mode-locked regions with bistable excitation edge. There is an even-all multimodal transition between the 2 : 1 and 3 : 1 states in the vicinity of the main resonance, which is analogous to the odd-all transition discovered earlier in the high-frequency regime. For I(0)<I(th) and small noise the firing rate has a maximum at the resonant frequency. For larger noise and subthreshold stimulation the maximum firing rate initially shifts toward lower frequencies, then returns to higher frequencies in the limit of large noise. The stochastic coherence antiresonance, defined as a simultaneous occurrence of (i) the maximum of the coefficient of variation and (ii) the minimum of the firing rate vs the noise intensity, occurs over a wide range of parameter values, including monostable regions. Results of this work can be verified experimentally.

Multi-frequency activation of neuronal networks by coordinated reset stimulation

We computationally study whether it is possible to stimulate a neuronal population in such a way that its mean firing rate increases without an increase of the population's net synchronization. For this, we use coordinated reset (CR) stimulation, which has previously been developed to desynchronize populations of oscillatory neurons. Intriguingly, delivered to a population of predominantly silent FitzHugh-Nagumo or Hindmarsh-Rose neurons at sufficient stimulation amplitudes, CR robustly causes a multi-frequency activation: different Arnold tongues such as 1 : 1 or n : m entrained neuronal clusters emerge, which consist of phase-shifted sub clusters. Owing to the clustering pattern the neurons' timing is well balanced, so that in total there is no synchronization. Our findings may contribute to the development of novel and safe stimulation treatments that specifically counteract cerebral hypo-activity without promoting pathological synchronization or inducing epileptic seizures.

Mode-locked spike trains in responses of ventral cochlear nucleus chopper and onset neurons to periodic stimuli

We report evidence of mode-locking to the envelope of a periodic stimulus in chopper units of the ventral cochlear nucleus (VCN). Mode-locking is a generalized description of how responses in periodically forced nonlinear systems can be closely linked to the input envelope, while showing temporal patterns of higher order than seen during pure phase-locking. Re-analyzing a previously unpublished dataset in response to amplitude modulated tones, we find that of 55% of cells (6/11) demonstrated stochastic mode-locking in response to sinusoidally amplitude modulated (SAM) pure tones at 50% modulation depth. At 100% modulation depth SAM, most units (3/4) showed mode-locking. We use interspike interval (ISI) scattergrams to unravel the temporal structure present in chopper mode-locked responses. These responses compared well to a leaky integrate-and-fire model (LIF) model of chopper units. Thus the timing of spikes in chopper unit responses to periodic stimuli can be understood in terms of the complex dynamics of periodically forced nonlinear systems. A larger set of onset (33) and chopper units (24) of the VCN also shows mode-locked responses to steady-state vowels and cosine-phase harmonic complexes. However, while 80% of chopper responses to complex stimuli meet our criterion for the presence of mode-locking, only 40% of onset cells show similar complex-modes of spike patterns. We found a correlation between a unit's regularity and its tendency to display mode-locked spike trains as well as a correlation in the number of spikes per cycle and the presence of complex-modes of spike patterns. These spiking patterns are sensitive to the envelope as well as the fundamental frequency of complex sounds, suggesting that complex cell dynamics may play a role in encoding periodic stimuli and envelopes in the VCN.

Cortical pyramidal cells as non-linear oscillators: experiment and spike-generation theory

Cortical neurons are capable of generating trains of action potentials in response to current injections. These discharges can take different forms, e.g., repetitive firing that adapts during the period of current injection or bursting behaviors. We have used a combined experimental and computational approach to characterize the dynamics leading to action potential responses in single neurons. Specifically we investigated the origin of complex firing patterns in response to sinusoidal current injections. Using a reduced model, the theta-neuron, alongside recordings from cortical pyramidal cells we show that both real and simulated neurons show phase-locking to sine wave stimuli up to a critical frequency, above which period skipping and 1-to-x phase-locking occurs. The locking behavior follows a complex "devil's staircase" phenomena, where locked modes are interleaved with irregular firing. We further show that the critical frequency depends on the time scale of spike generation and on the level of spike frequency adaptation. These results suggest that phase-locking of neuronal responses to complex input patterns can be explained by basic properties of the spike-generating machinery.

Continuation and bifurcation analysis of a periodically forced excitable system

The response of an excitable cell to periodic electrical stimulation is modeled using the FitzHugh-Nagumo (FHN) system submitted to a gaussian-shaped pacing, the width of which is small compared with the action potential duration. The influence of the amplitude and the period of the stimulation is studied using numerical continuation and bifurcation techniques (AUTO97 software). Results are discussed in the light of prior experimental and theoretical findings. In particular, agreement with the documented behavior of periodically stimulated cardiac cells and squid axons is discussed. As previously reported, we find many different "M:N" periodic solutions, period-doubling sequences leading to seemingly chaotic regimes, and bistability phenomena. In addition, the use of continuation techniques has allowed us to track unstable solutions of the system and thus to determine how the different stable rhythms are connected with each other in a bifurcation diagram. Depending on the stimulus amplitude, the aspect of the bifurcation diagram with the stimulus period as main varying parameter can vary from very simple to very complex. In its most developed structure, this bifurcation diagram consists of a main "tree" of period-2(P) branches, where the 1:1, 1:0, 2:2, 2:1,... rhythms are located, and of several closed loops made up of period-{N x 2(P)} branches (N>2), isolated from each other and from the main tree. It is mainly on such loops that N:1 rhythms (N>2) on one hand, and N:N-1 or Wenckebach rhythms (N>2) on the other hand, are located. Stable M:N and M:N-1 rhythms (M>or=N) can be found on the same branch of solutions. They are separated by a region of unstable solutions at small stimulus amplitudes, but this region shrinks gradually as the stimulus amplitude is raised, until it finally disappears. We believe that this property is related to the excitability characteristics of the FHN system. It would be interesting to know if it has any correspondence in the behavior of real excitable cells.

Bidirectional modulation of neuronal responses by depolarizing GABAergic inputs

The reversal potential of GABAA receptor channels is known to be less negative than the resting membrane potential under some cases. Recent electrophysiological experiments revealed that a GABAergic unitary conductance with such a depolarized reversal potential could not only prevent but also facilitate action potential generation depending on the timing of its application relative to the excitatory unitary conductance. Using a two-dimensional point neuron model, we simulate the experiments regarding the integration of unitary conductances, and execute bifurcation analysis. Then we extend our analysis to the case in which the neuron receives two kinds of periodic input trains-an excitatory one and a GABAergic one. We show that the periodic depolarizing GABAergic input train can modulate the output time-averaged firing rate bidirectionally, namely as an increase or a decrease, in a devil's-staircase-like manner depending on the phase difference with the excitatory input train. Bifurcation analysis reveals the existence of a wide variety of phase-locked solutions underlying such a graded response of the neuron. We examine how the input time-width and the value of the GABAA reversal potential affect the response. Moreover, considering a neuronal population, we show that depolarizing GABAergic inputs bidirectionally modulate the amplitude of the oscillatory population activity.

Spontaneous switching of frequency-locking by periodic stimulus in oscillators of plasmodium of the true slime mold

Microfabrication technique was used to construct a model system with a living cell of plasmodium of the true slime mold, Physarum polycephalum, a living coupled oscillator system. Its parameters can be systematically controlled as in computer simulations, so that results are directly comparable to those of general mathematical models. As the first step, we investigated responses in oscillatory cells, the oscillators of the plasmodium, to periodic stimuli by temperature changes to elucidate characteristics of the cells as nonlinear systems whose internal dynamics are unknown because of their complexity. We observed that the forced oscillator of the plasmodium show 1:1, 2:1, 3:1 frequency locking inside so-called Arnold tongues regions as well as in other nonlinear systems such as chemical systems and other biological systems. In addition, we found spontaneous switching behavior from certain frequency locking states to other states, even under certain fixed parameters. This technique can be applied to more complex systems with multiple elements, such as coupled oscillator systems, and would be useful to investigate complicated phenomena in biological systems such as information processing.

Reliability of spike timing is a general property of spiking model neurons

The responses of neurons to time-varying injected currents are reproducible on a trial-by-trial basis in vitro, but when a constant current is injected, small variances in interspike intervals across trials add up, eventually leading to a high variance in spike timing. It is unclear whether this difference is due to the nature of the input currents or the intrinsic properties of the neurons. Neuron responses can fail to be reproducible in two ways: dynamical noise can accumulate over time and lead to a desynchronization over trials, or several stable responses can exist, depending on the initial condition. Here we show, through simulations and theoretical considerations, that for a general class of spiking neuron models, which includes, in particular, the leaky integrate-and-fire model as well as nonlinear spiking models, aperiodic currents, contrary to periodic currents, induce reproducible responses, which are stable under noise, change in initial conditions and deterministic perturbations of the input. We provide a theoretical explanation for aperiodic currents that cross the threshold.

Firing pattern modulation by oscillatory input in supragranular pyramidal neurons

Neocortical neurons in vivo receive periodic stimuli due to feedforward input from the periphery as well as local cellular and circuit properties. In order to understand how neurons process such information, the responses of neurons to periodic sine wave current stimuli of varying frequencies and amplitudes were investigated. Sine wave stimuli were injected into pyramidal cells of young adult ferret visual cortical slices in vitro using sharp microelectrodes. To simulate higher resting membrane potentials observed in vivo a slight depolarizing current was injected to bring the neuron just to threshold. Initially, neurons discharged at least one action potential per sine wave cycle, but as the frequency was increased, a point was reached where this one-to-one responsiveness was lost. This critical frequency was dependent upon the injected sine wave amplitude and the magnitude of the underlying steady-state depolarization, and was correlated with spike width. Larger steady-state depolarizations and thinner action potentials corresponded to higher critical frequencies. Thus, when a neuron was very active it could respond in a one-to-one fashion over a greater range of frequencies than with the smallest DC offset. The results suggest that the frequency-following characteristics of individual cortical neurons can be modulated by the activity state of the neuron itself.

Responses of a Hodgkin-Huxley neuron to various types of spike-train inputs

Numerical investigations have been made of responses of a Hodgkin-Huxley (HH) neuron to spike-train inputs whose interspike interval (ISI) is modulated by deterministic, semi-deterministic (chaotic), and stochastic signals. As deterministic one, we adopt inputs with the time-independent ISI and with time-dependent ISI modulated by sinusoidal signal. The Rössler and Lorentz models are adopted for chaotic modulations of ISI. Stochastic ISI inputs with the gamma distribution are employed. It is shown that distribution of output ISI data depends not only on the mean of ISIs of spike-train inputs but also on their fluctuations. The distinction of responses to the three kinds of inputs can be made by return maps of input and output ISIs, but not by their histograms. The relation between the variations of input and output ISIs is shown to be different from that of the integrate and fire (IF) model because of the refractory period in the HH neuron.

Phase sensitivity and entrainment in a modeled bursting neuron

A model of neuron R15 in Aplysia was used to study the mechanisms determining the phase-response curve (PRC) of the cell in response to both extrinsic current pulses and modeled synaptic input and to compare entrainment predictions from PRCs with those from actual simulations. Over the range of stimulus parameters studied, the PRCs of the model exhibited minimal dependence upon stimulus amplitude, and a strong dependence upon stimulus duration. State-space analysis of the effect of transient current pulses provided several important insights into the relationship between the PRC and the underlying dynamics of the model, such as a correlation between the prestimulus concentration of Ca2+ and the poststimulus phase of the oscillation. The system nullclines were also found to provide well-defined limits upon the perturbatory extent of a hyperpolarizing input. These results demonstrated that experimentally applied current pulses are sufficient to determine the shape of the PRC in response to a synaptic input, provided that the duration of the current pulse is of a duration similar to that of the evoked synaptic current. Furthermore, we found that predictions of phase-locked 1:m entrainment from PRCs were valid, even when the duration of the periodically applied pulses were a significant portion of the control limit cycle.

The origins of aperiodicities in sensory neuron entrainment

Aperiodic entrainment to rhythmic sensory input was obtained with either a single neuron or an excitatory network model, without addition of a stochastic or "noisy" element. The entrainment properties of primary sensory neurons were well captured by the dynamics of the Hodgkin-Huxley ordinary differential equations with a quiescent resting state or threshold for spike output. The frequency-amplitude parameter space was compressed and aperiodic regimes were small in comparison to those of periodically activated pacemaker-like neurons. Transitions between phase-locked and aperiodic entrainment patterns were predictable and determined by the equation dynamics, supporting the contention that some aperiodicities observed in situ arise from the inherent membrane properties of neurons. When the rhythmically activated neuron was embedded in an excitatory network of Hodgkin-Huxley neurons with heterogeneous synaptic delays, aperiodic entrainment patterns were more frequently encountered and these were associated with asynchronous output from the network. Embedding the rhythmically activated neuron in a network with synaptic delays greatly reduced the range of entrained spike frequencies and increased the variability in the neuronal firing. The temporal coding of sensory stimuli may be dependent on these findings. Sensory stimuli are signaled in the periphery by a mixture of periodic and irregular interspike intervals. Most models of such temporal codes assume intrinsic rhythmicity arising from the ionic currents, with variations attributed to membrane or synaptic noise. In contrast, we demonstrate irregular neural codes that arise completely in the absence of noise. In the proposed model, the sources of these irregular sensory patterns are the extensive cross-connections and resultant interactions between neurons. The balance between the regular and irregular entrainment of a neuron in situ could uniquely identify a stimulus. Other biological mechanisms of modifying the entrainment properties and promoting aperiodic entrainment are discussed.

A note on modeling mechano-chemical transduction with an application to a skin receptor

Strain-sensitive (also called stretch-sensitive) ionic channels are thought to be present in various mechanoreceptors. The gating of these channels is precipitated by mechanical strains, as opposed to the usual activation processes of changes in membrane potential or other ligands. Below we present a class of models for the strain-activated mechanism, compare our approach to one of Sachs and Lecar (1991) and apply the gating mechanism to a model of a specific mechanoreceptor, namely a Pacinian corpuscle neurite model. The simulation experiment suggests the activation energy of the channel depends linearly, rather than nonlinearly, on the (hoop) strain in the receptor membrane.

Model of the dynamics of receptor potential in a mechanoreceptor

Mechanoreceptors are physiological units that convert mechanical stimuli to neural responses. An important one in the skin is the Pacinian corpuscle, which consists of a nerve ending surrounded by a capsule that is made up of alternating layers of fluid and elastic shells. In this paper a capsule model is coupled to a model for the dendrite via a variable representing the hoop strain imposed by the capsule on the receptor membrane of the dendrite. The transient behavior of the receptor potential due to step and periodic displacement-type stimuli imposed on the outer membrane of the capsule is investigated, and the results are compared to those of analogous experiments. The model is used to develop hypotheses concerning the role of the capsule as a filter for mechanical stimuli.

Intermittent chaos, self-organization, and learning from synchronous synaptic activity in model neuron networks

Self-organization of frequencies is studied by using model neurons called VCONs (voltage-controlled oscillator neuron models). These models give direct access to frequency information, in contrast to all-or-none neuron models, and they generate voltage spikes that phase-lock to oscillatory stimulation, similar to phase-locking of action potentials to oscillatory voltage stimulation observed in Hodgkin-Huxley preparations of squid axons. The rotation vector method is described and used to study how networks synchronize, even in the presence of noise or when damaged; the entropy of ratios of phases is used to construct an energy function that characterizes organized behavior. Computer simulations show that rotation numbers (output frequency/input frequency) describe both chaotic and nonchaotic behavior. Learning occurs when synaptic connections strengthen in response to stimulation that is synchronous with cell activity. It is shown that intermittent chaotic firing is suppressed and simple stable responses are enhanced by such learning in VCON networks. This analysis provides a rigorous basis for further investigation of the ideas of Wiener [Wiener, N. (1961) Cybernetics (MIT Press, Cambridge, MA), p. 191] on the origin of slow brain waves due to "the pulling together of frequencies."

Dynamic relationship between the slow potential and spikes in cockroach ocellar neurons

The relationship between the slow potential and spikes of second-order ocellar neurons of the cockroach, Periplaneta americana, was studied. The stimulus was a sinusoidally modulated light with various mean illuminances. A solitary spike was generated at the depolarizing phase of the modulation response. Analysis of the relationship between the amplitude/frequency of voltage modulation and the rate of spike generation showed that (a) the spike initiation process was bandpass at approximately 0.5-5 Hz, (b) the process contained a dynamic linearity and a static nonlinearity, and (c) the spike threshold at optimal frequencies (0.5-5 Hz) remained unchanged over a mean illuminance range of 3.6 log units, whereas (d) the spike threshold at frequencies of less than 0.5 Hz was lower at a dimmer mean illuminance. The voltage noise in the response was larger and the mean membrane potential level was more positive at a dimmer mean illuminance. Steady or noise current injection during sinusoidal light stimulation showed that (a) the decrease in the spike threshold at a dimmer mean illuminance was due to the increase in the noise variance: the noise had facilitatory effects on the spike initiation; and (b) the change in the mean potential level had little effect on the spike threshold. We conclude that fundamental signal modifications occur during the spike initiation in the cockroach ocellar neuron, a finding that differs from the spike initiation process in other visual systems, including Limulus eye and vertebrate retina, in which it is presumed that little signal modification occurs at the analog-to-digital conversion process.

Frequency modulation dynamics in neural networks

The VCON model described here shares certain qualitative features of stimulus-response characteristics with the data in FIGURE 1 and TABLE 1. At the same time, it provides an uncomplicated methodology for modelling neural networks that emphasizes their frequency aspects. In particular, models based on VCONs are amenable to the rotation vector method, which can be used to uncover stable synchronization of firing within a network. This is described in the appendix, where it is shown that the stable firing patterns within the network correspond to minima of an associated (local) energy function. We have seen here how a model of a simple CPG for breathing can be constructed and analyzed. Similar models for rhythm splitting of small mammal activity cycles, sound location networks, and motility in the gastrointestinal tract have been constructed, and large networks of VCONs have been shown to have stable spatial patterns of synchronization.

Chaos in biological systems

Chaos is a widespread and easily recognizable phenomenon that hardly anybody took notice of until recently. The reason may be that chaos has something profoundly counterintuitive about it. It will not fit easily into any familiar cause–effect frame. The best introduction to chaos is by the way of an example. Consider a leaking faucet (Shaw, 1984). When the weight of the accumulating drop exceeds the surface tension the drop falls and a new drop begins to form. If the leak is small and the pressure in the faucet is constant, the time taken for the drop to reach the critical weight is constant. The dripping is perfectly periodic, the period depending on the leak rate. If the leak is slightly increased, the period of dripping will decrease slightly and vice versa. However, somewhere beyond this point the leaking faucet becomes a nuisance. When the leak is increased beyond a certain point the dripping looses its regularity. The time interval between the drops will first alternate periodically between a short and a long time interval. After a further increase of the leak this double periodic pattern will become unstable and change into a new pattern where four different time intervals between the drops alternate periodically. As the leak is further increased the period will double again and again and finally the dripping becomes completely irregular without any repeating pattern. When this occurs we are observing chaos. At the same time we are posed with the problem of understanding how such a ridiculously simple system can show random behaviour.

Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Membrane potential responses of a Hodgkin-Huxley oscillator to an externally-applied sinusoidal current were numerically calculated with relation to bifurcation parameters of the amplitude and the frequency of the stimulating current. The Hodgkin-Huxley oscillator, or the Hodgkin-Huxley axon in the state of self-sustained oscillation of action potentials, was realized by immersing the axon in calcium-deficient sea water. The forced oscillations were analysed by the stroboscopic plots and/or the Lorenz plots. The results show that the periodically forced Hodgkin-Huxley oscillator exhibits not only periodic motions (harmonic or sub-harmonic synchronization) but also non-periodic motions (quasi-periodic or chaotic oscillation), that the motions were determined by the amplitude and the frequency of the stimulating current, and that the characteristic motions obtained in the present study were in reasonable agreement with those of our previous results, found experimentally in squid giant axons. Also, two kinds of routes to the chaotic oscillations were found; successive period-doubling bifurcations and formation of the intermittently chaotic oscillation from sub-harmonic synchronization.

Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: a theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias

A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977, 1980), large perturbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincaré or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincaré map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincaré map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrioventricular (AV) block clinically observed. In particular 2:2 and 4:2 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics.

Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells

The spontaneous rhythmic activity of aggregates of embryonic chick heart cells was perturbed by the injection of single current pulses and periodic trains of current pulses. The regular and irregular dynamics produced by periodic stimulation were predicted theoretically from a mathematical analysis of the response to single pulses. Period-doubling bifurcations, in which the period of a regular oscillation doubles, were predicted theoretically and observed experimentally.

Autorhythmicity and entrainment in excitable membranes

Low calcium increases the excitability of neurones and can induce autorhythmicity in excitable cells. Numerical solutions of the Hodgkin-Huxley membrane equations, and numerical evaluations of the small-signal impedance and admittance are used to illustrate the increase in resonance produced by low [Ca2+]0. The resonant frequency may be located either by the peak of the amplitude of the impedance, or by the frequency at which the phase angle is zero for 1 : 1 entrained action potentials. Autorhythmicity is produced by any mechanism which increases the resonant peak of the amplitude of the membrane impedance.