Slow Resting State Fluctuations Enhance Neuronal and Behavioral Responses to Looming Sounds.

We investigate both experimentally and using a computational model how the power of the electroencephalogram (EEG) recorded in human subjects tracks the presentation of sounds with acoustic intensities that increase exponentially (looming) or remain constant (flat). We focus on the link between this EEG tracking response, behavioral reaction times and the time scale of fluctuations in the resting state, which show considerable inter-subject variability. Looming sounds are shown to generally elicit a sustained power increase in the alpha and beta frequency bands. In contrast, flat sounds only elicit a transient upsurge at frequencies ranging from 7 to 45 Hz. Likewise, reaction times (RTs) in an audio-tactile task at different latencies from sound onset also present significant differences between sound types. RTs decrease with increasing looming intensities, i.e. as the sense of urgency increases, but remain constant with stationary flat intensities. We define the reaction time variation or "gain" during looming sound presentation, and show that higher RT gains are associated with stronger correlations between EEG power responses and sound intensity. Higher RT gain further entails higher relative power differences between loom and flat in the alpha and beta bands. The full-width-at-half-maximum of the autocorrelation function of the eyes-closed resting state EEG also increases with RT gain. The effects are topographically located over the central and frontal electrodes. A computational model reveals that the increase in stimulus-response correlation in subjects with slower resting state fluctuations is expected when EEG power fluctuations at each electrode and in a given band are viewed as simple coupled low-pass filtered noise processes jointly driven by the sound intensity. The model assumes that the strength of stimulus-power coupling is proportional to RT gain in different coupling scenarios, suggesting a mechanism by which slower resting state fluctuations enhance EEG response and shorten reaction times.

Cooling reverses pathological bifurcations to spontaneous firing caused by mild traumatic injury.

Mild traumatic injury can modify the key sodium (Na+) current underlying the excitability of neurons. It causes the activation and inactivation properties of this current to become shifted to more negative trans-membrane voltages. This so-called coupled left shift (CLS) leads to a chronic influx of Na+ into the cell that eventually causes spontaneous or "ectopic" firing along the axon, even in the absence of stimuli. The bifurcations underlying this enhanced excitability have been worked out in full ionic models of this effect. Here, we present computational evidence that increased temperature T can exacerbate this pathological state. Conversely, and perhaps of clinical relevance, mild cooling is shown to move the naturally quiescent cell further away from the threshold of ectopic behavior. The origin of this stabilization-by-cooling effect is analyzed by knocking in and knocking out, one at a time, various processes thought to be T-dependent. The T-dependence of the Na+ current, quantified by its Q 10-Na factor, has the biggest impact on the threshold, followed by Q 10-pump of the sodium-potassium exchanger. Below the ectopic boundary, the steady state for the gating variables and the resting potential are not modified by temperature, since our model separately tallies the Na+ and K+ ions including their separate leaks through the pump. When only the gating kinetics are considered, cooling is detrimental, but in the full T-dependent model, it is beneficial because the other processes dominate. Cooling decreases the pump's activity, and since the pump hyperpolarizes, less hyperpolarization should lead to more excitability and ectopic behavior. But actually the opposite happens in the full model because decreased pump activity leads to smaller gradients of Na+ and K+, which in turn decreases the driving force of the Na+ current.

Modeling cancelation of periodic inputs with burst-STDP and feedback.

Prediction and cancelation of redundant information is an important feature that many neural systems must display in order to efficiently code external signals. We develop an analytic framework for such cancelation in sensory neurons produced by a cerebellar-like structure in wave-type electric fish. Our biologically plausible mechanism is motivated by experimental evidence of cancelation of periodic input arising from the proximity of conspecifics as well as tail motion. This mechanism involves elements present in a wide range of systems: (1) stimulus-driven feedback to the neurons acting as detectors, (2) a large variety of temporal delays in the pathways transmitting such feedback, responsible for producing frequency channels, and (3) burst-induced long-term plasticity. The bursting arises from back-propagating action potentials. Bursting events drive the input frequency-dependent learning rule, which in turn affects the feedback input and thus the burst rate. We show how the mean firing rate and the rate of production of 2- and 4-spike bursts (the main learning events) can be estimated analytically for a leaky integrate-and-fire model driven by (slow) sinusoidal, back-propagating and feedback inputs as well as rectified filtered noise. The effect of bursts on the average synaptic strength is also derived. Our results shed light on why bursts rather than single spikes can drive learning in such networks "online", i.e. in the absence of a correlative discharge. Phase locked spiking in frequency specific channels together with a frequency-dependent STDP window size regulate burst probability and duration self-consistently to implement cancelation.

Optimal heterogeneity for coding in spiking neural networks.

The effect of cellular heterogeneity on the coding properties of neural populations is studied analytically and numerically. We find that heterogeneity decreases the threshold for synchronization, and its strength is nonlinearly related to the network mean firing rate. In addition, conditions are shown under which heterogeneity optimizes network information transmission for either temporal or rate coding, with high input frequencies leading to different effects for each coding strategy. The results are shown to be robust for more realistic conditions.

Oscillatory response in a sensory network of ON and OFF cells with instantaneous and delayed recurrent connections.

A neural field model with multiple cell-to-cell feedback connections is investigated. Our model incorporates populations of ON and OFF cells, receiving sensory inputs with direct and inverted polarity, respectively. Oscillatory responses to spatially localized stimuli are found to occur via Andronov-Hopf bifurcations of stationary activity. We explore the impact of multiple delayed feedback components as well as additional excitatory and/or inhibitory non-delayed recurrent signals on the instability threshold. Paradoxically, instantaneous excitatory recurrent terms are found to enhance network responsiveness by reducing the oscillatory response threshold, allowing smaller inputs to trigger oscillatory activity. Instantaneous inhibitory components do the opposite. The frequency of these response oscillations is further shaped by the polarity of the non-delayed terms.

Pulse-coupled neuron models as investigative tools for musical consonance.

We investigate the mode locking properties of simple dynamical models of pulse-coupled neurons to two tones, i.e., simple musical intervals. A recently proposed nonlinear synchronization theory of musical consonance links the subjective ranking from consonant to dissonant intervals to the universal ordering of robustness of mode locking ratios in forced nonlinear oscillators. The theory was illustrated using two leaky integrate-and-fire neuron models with mutual excitatory coupling, with each neuron firing at one of the two frequencies in the musical interval. We show that the ordering of mode locked states in such models is not universal, but depends on coupling strength. Further, unless the coupling is weak, the observed ratio of firing frequencies is higher than that of the input tones. We finally explore generic aspects of a possible synchronization theory by driving the model neurons with sinusoidal forcing, leading to down-converted, more realistic firing rates. This model exhibits one-to-one entrainment when the input frequencies are in simple ratios. We also consider the robustness to the presence of noise that is present in the neural firing activity. We briefly discuss agreements and discrepancies between predictions from this theory and physiological/psychophysical data, and suggest directions in which to develop this theory further.

Envelope gating and noise shaping in populations of noisy neurons.

Narrowband signals have fast and slow time scales. The transmission of narrowband signal features on both times cales, by spiking neurons, is demonstrated experimentally and theoretically. The interaction of the narrowband input and the threshold nonlinearity may create out-of-band interference, hindering the transmission of signals in a low-frequency range. The resultant out-of-band signal is the "envelope," or time-varying modulation of the narrowband signal. The levels of noise and nonlinearity intrinsic to the neuron gate transmission on the slow "envelope" time scale. When a narrowband and a distinct slow signal drive the neuron, the slow signal may be poorly transmitted. Increasing intrinsic noise in an averaging network removes the envelope in favor of the slow signal, paradoxically increasing the signal-to-noise ratio. These gating effects are generic for threshold and excitable systems.

Firing statistics of a neuron model driven by long-range correlated noise.

We study the statistics of the firing patterns of a perfect integrate and fire neuron model driven by additive long-range correlated Ornstein-Uhlenbeck noise. Using a quasistatic weak noise approximation we obtain expressions for the interspike interval (ISI) probability density, the power spectral density, and the spike count Fano factor. We find unimodal, long-tailed ISI densities, Lorenzian power spectra at low frequencies, and a minimum in the Fano factor as a function of counting time. The implications of these results for signal detection are discussed.

Two-category model of task allocation with application to ant societies.

In many network models of interacting units such as cells or insects, the coupling coefficients between units are independent of the state of the units. Here we analyze the temporal behavior of units that can switch between two 'category' states according to rules that involve category-dependent coupling coefficients. The behaviors of the category populations resulting from the asynchronous random updating of units are first classified according to the signs of the coupling coefficients using numerical simulations. They range from isolated fixed points to lines of fixed points and stochastic attractors. These behaviors are then explained analytically using iterated function systems and birth-death jump processes. The main inspiration for our work comes from studies of non-hierarchical task allocation in, e.g., harvester ant colonies where temporal fluctuations in the numbers of ants engaged in various tasks occur as circumstances require and depend on interactions between ants. We identify interaction types that produce quick recovery from perturbations to an asymptotic behavior whose characteristics are function of the coupling coefficients between ants as well as between ants and their environment. We also compute analytically the probability density of the population numbers, and show that perturbations in our model decay twice as fast as in a model with random switching dynamics. A subset of the interaction types between ants yields intrinsic stochastic asymptotic behaviors which could account for some of the experimentally observed fluctuations. Such noisy trajectories are shown to be random walks with state-dependent biases in the 'category population' phase space. With an external stimulus, the parameters of the category-switching rules become time-dependent. Depending on the growth rate of the stimulus in comparison to its population-dependent decay rate, the dynamics may qualitatively differ from the case without stimulus. Our simple two-category model provides a framework for understanding the rich variety of behaviors in network dynamics with state-dependent coupling coefficients, and especially in task allocation processes with many tasks.

Model of gamma frequency burst discharge generated by conditional backpropagation.

Pyramidal cells of the electrosensory lateral line lobe (ELL) of the weakly electric fish Apteronotus leptorhynchus have been shown to produce oscillatory burst discharge in the gamma-frequency range (20-80 Hz) in response to constant depolarizing stimuli. Previous in vitro studies have shown that these bursts arise through a recurring spike backpropagation from soma to apical dendrites that is conditional on the frequency of action potential discharge ("conditional backpropagation"). Spike bursts are characterized by a progressive decrease in inter-spike intervals (ISIs), and an increase of dendritic spike duration and the amplitude of a somatic depolarizing afterpotential (DAP). The bursts are terminated when a high-frequency somatic spike doublet exceeds the dendritic spike refractory period, preventing spike backpropagation. We present a detailed multi-compartmental model of an ELL basilar pyramidal cell to simulate somatic and dendritic spike discharge and test the conditions necessary to produce a burst output. The model ionic channels are described by modified Hodgkin-Huxley equations and distributed over both soma and dendrites under the constraint of available immunocytochemical and electrophysiological data. The currents modeled are somatic and dendritic sodium and potassium involved in action potential generation, somatic and proximal apical dendritic persistent sodium, and K(V)3.3 and fast transient A-like potassium channels distributed over the entire model cell. The core model produces realistic somatic and dendritic spikes, differential spike refractory periods, and a somatic DAP. However, the core model does not produce oscillatory spike bursts with constant depolarizing stimuli. We find that a cumulative inactivation of potassium channels underlying dendritic spike repolarization is a necessary condition for the model to produce a sustained gamma-frequency burst pattern matching experimental results. This cumulative inactivation accounts for a frequency-dependent broadening of dendritic spikes and results in a conditional failure of backpropagation when the intraburst ISI exceeds dendritic spike refractory period, terminating the burst. These findings implicate ion channels involved in repolarizing dendritic spikes as being central to the process of conditional backpropagation and oscillatory burst discharge in this principal sensory output neuron of the ELL.

Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli.

Accurate detection of sensory input is essential for the survival of a species. Weakly electric fish use amplitude modulations of their self-generated electric field to probe their environment. P-type electroreceptors convert these modulations into trains of action potentials. Cumulative relative refractoriness in these afferents leads to negatively correlated successive interspike intervals (ISIs). We use simple and accurate models of P-unit firing to show that these refractory effects lead to a substantial increase in the animal's ability to detect sensory stimuli. This assessment is based on two approaches, signal detection theory and information theory. The former is appropriate for low-frequency stimuli, and the latter for high-frequency stimuli. For low frequencies, we find that signal detection is dependent on differences in mean firing rate and is optimal for a counting time at which spike train variability is minimal. Furthermore, we demonstrate that this minimum arises from the presence of negative ISI correlations at short lags and of positive ISI correlations that extend out to long lags. Although ISI correlations might be expected to reduce information transfer, in fact we find that they improve information transmission about time-varying stimuli. This is attributable to the differential effect that these correlations have on the noise and baseline entropies. Furthermore, the gain in information transmission rate attributable to correlations exhibits a resonance as a function of stimulus bandwidth; the maximum occurs when the inverse of the cutoff frequency of the stimulus is of the order of the decay time constant of refractory effects. Finally, we show that the loss of potential information caused by a decrease in spike-timing resolution is smaller for low stimulus cutoff frequencies than for high ones. This suggests that a rate code is used for the encoding of low-frequency stimuli, whereas spike timing is important for the encoding of high-frequency stimuli.

Variability of the electric organ discharge interval duration in resting Gymnotus carapo.

We recorded the electric organ discharges of resting Gymnotus carapo specimens. We analyzed the time series formed by the sequence of interdischarge intervals. Nonlinear prediction, false nearest neighbor analyses, and comparison between the performance of nonlinear and linear autoregressive models fitted to the data indicated that nonlinear correlations between intervals were absent, or were present to a minor extent only. Following these analyses, we showed that linear autoregressive models with combined Gaussian and shot noise reproduced the variability and correlations of the resting discharge pattern. We discuss the implications of our findings for the mechanisms underlying the timing of electric organ discharge generation. We also argue that autoregressive models can be used to evaluate the changes arising during a wide variety of behaviors, such as the modification in the discharge intervals during interaction between fish pairs.

Subtractive and divisive inhibition: effect of voltage-dependent inhibitory conductances and noise.

The influence of voltage-dependent inhibitory conductances on firing rate versus input current (f-I) curves is studied using simulations from a new compartmental model of a pyramidal cell of the weakly electric fish Apteronotus leptorhynchus. The voltage dependence of shunting-type inhibition enhances the subtractive effect of inhibition on f-I curves previously demonstrated in Holt and Koch (1997) for the voltage-independent case. This increased effectiveness is explained using the behavior of the average subthreshold voltage with input current and, in particular, the nonlinearity of Ohm's law in the subthreshold regime. Our simulations also reveal, for both voltage-dependent and -independent inhibitory conductances, a divisive inhibition regime at low frequencies (f < 40 Hz). This regime, dependent on stochastic inhibitory synaptic input and a coupling of inhibitory strength and variance, gives way to subtractive inhibition at higher-output frequencies (f > 40 Hz). A simple leaky integrate-and-fire type model that incorporates the voltage dependence supports the results from our full ionic simulations.

Rate processes in a delayed, stochastically driven, and overdamped system

A Fokker-Planck formulation of systems described by stochastic delay differential equations has been recently proposed. A separation of time scales approximation allowing this Fokker-Planck equation to be simplified in the case of multistable systems is hereby introduced, and applied to a system consisting of a particle coupled to a delayed quartic potential. In that approximation, population numbers in each well obey a phenomenological rate law. The corresponding transition rate is expressed in terms of the noise variance and the steady-state probability density. The same type of expression is also obtained for the mean first passage time from a given point to another one. The steady-state probability density appearing in these formulas is determined both from simulations and from a small delay expansion. The results support the validity of the separation of time scales approximation. However, the results obtained using a numerically determined steady-state probability are more accurate than those obtained using the small delay expansion, thereby stressing the high sensitivity of the transition rate and mean first passage time to the shape of the steady-state probability density. Simulation results also indicate that the transition rate and the mean first passage time both follow Arrhenius' law when the noise variance is small, even if the delay is large. Finally, deterministic unbounded solutions are found to coexist with the bounded ones. In the presence of noise, the transition rate from bounded to unbounded solutions increases with the delay.

Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors.

Weakly electric fish generate a periodic electric field as a carrier signal for active location and communication tasks. Highly sensitive P-type receptors on their surface fire in response to carrier amplitude modulations (AM's) in a noisy phase locked fashion. A simple generic model of receptor activity and signal encoding is presented. Its suprathreshold dynamics, memory and receptor noise reproduce observed firing interval distributions and correlations. The model ultimately explains how smooth responses to AM's are compatible with its nonlinear phase locking properties, and reveals how receptor noise can sometimes enhance the encoding of small yet suprathreshold AM's.

A phase model of temperature-dependent mammalian cold receptors.

We present a tractable stochastic phase model of the temperature sensitivity of a mammalian cold receptor. Using simple linear dependencies of the amplitude, frequency, and bias on temperature, the model reproduces the experimentally observed transitions between bursting, beating, and stochastically phase-locked firing patterns. We analyze the model in the deterministic limit and predict, using a Strutt map, the number of spikes per burst for a given temperature. The inclusion of noise produces a variable number of spikes per burst and also extends the dynamic range of the neuron, both of which are analyzed in terms of the Strutt map. Our analysis can be readily applied to other receptors that display various bursting patterns following temperature changes.

Spike train patterning and forecastability.

Theories of neural coding rely on a knowledge of correlations between firing events. These correlations are also useful to validate biophysical models for the neural activity. We present a methodology for validating models based on the assessment of linear and non-linear correlations between variables derived from the spike train. The firing pattern of an electroreceptor is analyzed in this framework. We show that a purely stochastic model fails to capture the essential correlations between interspike intervals, even though it reproduces the interval histogram and certain spike train spectral features. However, a biophysical model, based on the Fitzhugh-Nagumo equations with noise, does exhibit many of the correlations seen in the data, including those between successive firing phases.

Encoding with bursting, subthreshold oscillations, and noise in mammalian cold receptors.

Mammalian cold thermoreceptors encode steady-state temperatures into characteristic temporal patterns of action potentials. We propose a mechanism for the encoding process. It is based on Plant's ionic model of slow wave bursting, to which stochastic forcing is added. The model reproduces firing patterns from cat lingual cold receptors as the parameters most likely to underlie the thermosensitivity of these receptors varied over a 25 degrees C range. The sequence of firing patterns goes from regular bursting, to simple periodic, to stochastically phase-locked firing or "skipping." The skipping at higher temperatures is shown to necessitate an interaction between noise and a subthreshold endogenous oscillation in the receptor. The basic period of all patterns is robust to noise. Further, noise extends the range of encodable stimuli. An increase in firing irregularity with temperature also results from the loss of stability accompanying the approach by the slow dynamics of a reverse Hopf bifurcation. The results are not dependent on the precise details of the Plant model, but are generic features of models where an autonomous slow wave arises through a Hopf bifurcation. The model also addresses the variability of the firing patterns across fibers. An alternate model of slow-wave bursting (Chay and Fan 1993) in which skipping can occur without noise is also analyzed here in the context of cold thermoreception. Our study quantifies the possible origins and relative contribution of deterministic and stochastic dynamics to the coding scheme. Implications of our findings for sensory coding are discussed.

Bistability and the dynamics of periodically forced sensory neurons.

Many neurons at the sensory periphery receive periodic input, and their activity exhibits entrainment to this input in the form of a preferred phase for firing. This article describes a modeling study of neurons which skip a random number of cycles of the stimulus between firings over a large range of input intensities. This behavior was investigated using analog and digital simulations of the motion of a particle in a double-well with noise and sinusoidal forcing. Well residence-time distributions were found to exhibit the main features of the interspike interval histograms (ISIH) measured on real sensory neurons. The conditions under which it is useful to view neurons as simple bistable systems subject to noise are examined by identifying the features of the data which are expected to arise for such systems. This approach is complementary to previous studies of such data based, e.g., on non-homogeneous point processes. Apart from looking at models which form the backbone of excitable models, our work allows us to speculate on the role that stochastic resonance, which can arise in this context, may play in the transmission of sensory information.