To determine why elements of central pattern generators phase lock in a particular pattern under some conditions but not others, we tested a theoretical pattern prediction method. The method is based on the tabulated open loop pulsatile interactions of bursting neurons on a cycle-by-cycle basis and was tested in closed loop hybrid circuits composed of one bursting biological neuron and one bursting model neuron coupled using the dynamic clamp. A total of 164 hybrid networks were formed by varying the synaptic conductances. The prediction of 1:1 phase locking agreed qualitatively with the experimental observations, except in three hybrid circuits in which 1:1 locking was predicted but not observed. Correct predictions sometimes required consideration of the second order phase resetting, which measures the change in the timing of the second burst after the perturbation. The method was robust to offsets between the initiation of bursting in the presynaptic neuron and the activation of the synaptic coupling with the postsynaptic neuron. The quantitative accuracy of the predictions fell within the variability (10%) in the experimentally observed intrinsic period and phase resetting curve (PRC), despite changes in the burst duration of the neurons between open and closed loop conditions.

In most species, the capability of perceiving and using the passage of time in the seconds-to-minutes range (interval timing) is not only accurate but also scalar: errors in time estimation are linearly related to the estimated duration. The ubiquity of scalar timing extends over behavioral, lesion, and pharmacological manipulations. For example, in mammals, dopaminergic drugs induce an immediate, scalar change in the perceived time (clock pattern), whereas cholinergic drugs induce a gradual, scalar change in perceived time (memory pattern). How do these properties emerge from unreliable, noisy neurons firing in the milliseconds range? Neurobiological information relative to the brain circuits involved in interval timing provide support for an striatal beat frequency (SBF) model, in which time is coded by the coincidental activation of striatal spiny neurons by cortical neural oscillators. While biologically plausible, the impracticality of perfect oscillators, or their lack thereof, questions this mechanism in a brain with noisy neurons. We explored the computational mechanisms required for the clock and memory patterns in an SBF model with biophysically realistic and noisy Morris–Lecar neurons (SBF–ML). Under the assumption that dopaminergic drugs modulate the firing frequency of cortical oscillators, and that cholinergic drugs modulate the memory representation of the criterion time, we show that our SBF–ML model can reproduce the pharmacological clock and memory patterns observed in the literature. Numerical results also indicate that parameter variability (noise) – which is ubiquitous in the form of small fluctuations in the intrinsic frequencies of neural oscillators within and between trials, and in the errors in recording/retrieving stored information related to criterion time – seems to be critical for the time-scale invariance of the clock and memory patterns.

In most species, interval timing is time-scale invariant: errors in time estimation scale up linearly with the estimated duration. In mammals, time-scale invariance is ubiquitous over behavioral, lesion, and pharmacological manipulations. For example, dopaminergic drugs induce an immediate, whereas cholinergic drugs induce a gradual, scalar change in timing. Behavioral theories posit that time-scale invariance derives from particular computations, rules, or coding schemes. In contrast, we discuss a simple neural circuit, the perceptron, whose output neurons fire in a clockwise fashion (interval timing) based on the pattern of coincidental activation of its input neurons. We show numerically that time-scale invariance emerges spontaneously in a perceptron with realistic neurons, in the presence of noise. Under the assumption that dopaminergic drugs modulate the firing of input neurons, and that cholinergic drugs modulate the memory representation of the criterion time, we show that a perceptron with realistic neurons reproduces the pharmacological clock and memory patterns, and their time-scale invariance, in the presence of noise. These results suggest that rather than being a signature of higher-order cognitive processes or specific computations related to timing, time-scale invariance may spontaneously emerge in a massively-connected brain from the intrinsic noise of neurons and circuits, thus providing the simplest explanation for the ubiquity of scale invariance of interval timing.

We used optogenetic mice to investigate possible nonlinear responses of the medial prefrontal cortex (mPFC) local network to light stimuli delivered by a 473 nm laser through a fiber optics. Every 2 s, a brief 10 ms light pulse was applied and the local field potentials (LFPs) were recorded with a 10 kHz sampling rate. The experiment was repeated 100 times and we only retained and analyzed data from six animals that showed stable and repeatable response to optical stimulations. The presence of nonlinearity in our data was checked using the null hypothesis that the data were linearly correlated in the temporal domain, but were random otherwise. For each trail, 100 surrogate data sets were generated and both time reversal asymmetry and false nearest neighbor (FNN) were used as discriminating statistics for the null hypothesis. We found that nonlinearity is present in all LFP data. The first 0.5 s of each 2 s LFP recording were dominated by the transient response of the networks. For each trial, we used the last 1.5 s of steady activity to measure the phase resetting induced by the brief 10 ms light stimulus. After correcting the LFPs for the effect of phase resetting, additional preprocessing was carried out using dendrograms to identify “similar” groups among LFP trials. We found that the steady dynamics of mPFC in response to light stimuli could be reconstructed in a three-dimensional phase space with topologically similar “8”-shaped attractors across different animals. Our results also open the possibility of designing a low-dimensional model for optical stimulation of the mPFC local network.

Understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple, and sometimes overlapping, inputs. If we are to use phase-resetting methods to analyze these circuits, we can either generate phase-resetting curves (PRCs) for all possible inputs and combinations of inputs, or we can develop an understanding of how to construct PRCs for arbitrary perturbations of a given neuron. The latter strategy is the goal of this study. We present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses. A geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space. The perturbations in current are treated as displacements in the direction corresponding to membrane voltage. We show that for type I oscillators, the direction of a perturbation in current is nearly tangent to the limit cycle; hence, the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting. In order to obtain the phase resetting in terms of elapsed time or temporal phase, a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion. This mapping is shown to be an invariant of the dynamics. Perturbations in current applied to type II oscillators produce significant normal displacements from the limit cycle, so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account. Empirical attempts to correct for differences in angular velocity (amplitude versus phase effects in terms of a circular coordinate system) during relaxation back to the limit cycle achieved some success in the construction of phase-resetting curves for type II model oscillators. The ultimate goal of this work is the extension of these techniques to biological circuits comprising type II neural oscillators, which appear frequently in identified central pattern-generating circuits.

Cognitive processes such as decision-making, rate calculation and planning require an accurate estimation of durations in the supra-second range—interval timing. In addition to being accurate, interval timing is scale invariant: the time-estimation errors are proportional to the estimated duration. The origin and mechanisms of this fundamental property are unknown. We discuss the computational properties of a circuit consisting of a large number of (input) neural oscillators projecting on a small number of (output) coincidence detector neurons, which allows time to be coded by the pattern of coincidental activation of its inputs. We showed analytically and checked numerically that time-scale invariance emerges from the neural noise. In particular, we found that errors or noise during storing or retrieving information regarding the memorized criterion time produce symmetric, Gaussian-like output whose width increases linearly with the criterion time. In contrast, frequency variability produces an asymmetric, long-tailed Gaussian-like output, that also obeys scale invariant property. In this architecture, time-scale invariance depends neither on the details of the input population, nor on the distribution probability of noise.

Blocking the small-conductance (SK) calcium-activated potassium channel promotes burst firing in dopamine neurons both in vivo and in vitro. In vitro, the bursting is unusual in that spiking persists during the hyperpolarized trough and frequently terminates by depolarization block during the plateau. We focus on the underlying plateau potential oscillation generated in the presence of both apamin and TTX, so that action potentials are not considered. We find that although the plateau potentials are mediated by a voltage-gated Ca(2+) current, they do not depend on the accumulation of cytosolic Ca(2+), then use a computational model to test the hypothesis that the slowly voltage-activated ether-a-go-go-related gene (ERG) potassium current repolarizes the plateaus. The model, which includes a material balance on calcium, is able to reproduce the time course of both membrane potential and somatic calcium concentration, and can also mimic the induction of plateau potentials by the calcium chelator BAPTA. The principle of separation of timescales was used to gain insight into the mechanisms of oscillation and its modulation using nullclines in the phase space. The model predicts that the plateau will be elongated and ultimately result in a persistent depolarization as the ERG current is reduced. This study suggests that the ERG current may play a role in burst termination and the relief of depolarization block in vivo.

Recordings of the membrane potential from a bursting neuron were used to reconstruct the phase curve for that neuron for a limited set of perturbations. These perturbations were inhibitory synaptic conductance pulses able to shift the membrane potential below the most hyperpolarized level attained in the free running mode. The extraction of the phase resetting curve from such a one-dimensional time series requires reconstruction of the periodic activity in the form of a limit cycle attractor. Resetting was found to have two components. In the first component, if the pulse was applied during a burst, the burst was truncated, and the time until the next burst was shortened in a manner predicted by movement normal to the limit cycle. By movement normal to the limit cycle, we mean a switch between two well-defined solution branches of a relaxation-like oscillator in a hysteretic manner enabled by the existence of a singular dominant slow process (variable). In the second component, the onset of the burst was delayed until the end of the hyperpolarizing pulse. Thus, for the pulse amplitudes we studied, resetting was independent of amplitude but increased linearly with pulse duration. The predicted and the experimental phase resetting curves for a pyloric dilator neuron show satisfactory agreement. The method was applied to only one pulse per cycle, but our results suggest it could easily be generalized to accommodate multiple inputs.