In a variety of neurons, action potentials (APs) initiate at the proximal axon, within a region called the axon initial segment (AIS), which has a high density of voltage-gated sodium channels (NaVs) on its membrane. In pyramidal neurons, the proximal AIS has been reported to exhibit a higher proportion of NaVs with gating properties that are "right-shifted" to more depolarized voltages, compared to the distal AIS. Further, recent experiments have revealed that as neurons develop, the spatial distribution of NaV subtypes along the AIS can change substantially, suggesting that neurons tune their excitability by modifying said distribution. When neurons are stimulated axonally, computational modelling has shown that this spatial separation of gating properties in the AIS enhances the backpropagation of APs into the dendrites. In contrast, in the more natural scenario of somatic stimulation, our simulations show that the same distribution can impede backpropagation, suggesting that the choice of orthodromic versus antidromic stimulation can bias or even invert experimental findings regarding the role of NaV subtypes in the AIS. We implemented a range of hypothetical NaV distributions in the AIS of three multicompartmental pyramidal cell models and investigated the precise kinetic mechanisms underlying such effects, as the spatial distribution of NaV subtypes is varied. With axonal stimulation, proximal NaV availability dominates, such that concentrating right-shifted NaVs in the proximal AIS promotes backpropagation. However, with somatic stimulation, the models are insensitive to availability kinetics. Instead, the higher activation threshold of right-shifted NaVs in the AIS impedes backpropagation. Therefore, recently observed developmental changes to the spatial separation and relative proportions of NaV1.2 and NaV1.6 in the AIS differentially impact activation and availability. The observed effects on backpropagation, and potentially learning via its putative role in synaptic plasticity (e.g. through spike-timing-dependent plasticity), are opposite for orthodromic versus antidromic stimulation, which should inform hypotheses about the impact of the developmentally regulated subcellular localization of these NaV subtypes.

Spike trains with negative interspike interval (ISI) correlations, in which long/short ISIs are more likely followed by short/long ISIs, are common in many neurons. They can be described by stochastic models with a spike-triggered adaptation variable. We analyze a phenomenon in these models where such statistically dependent ISI sequences arise in tandem with quasi-statistically independent and identically distributed (quasi-IID) adaptation variable sequences. The sequences of adaptation states and resulting ISIs are linked by a nonlinear decorrelating transformation. We establish general conditions on a family of stochastic spiking models that guarantee this quasi-IID property and establish bounds on the resulting baseline ISI correlations. Inputs that elicit weak firing rate changes in samples with many spikes are known to be more detectible when negative ISI correlations are present because they reduce spike count variance; this defines a variance-reduced firing rate coding benchmark. We performed a Fisher information analysis on these adapting models exhibiting ISI correlations to show that a spike pattern code based on the quasi-IID property achieves the upper bound of detection performance, surpassing rate codes with the same mean rate-including the variance-reduced rate code benchmark-by 20% to 30%. The information loss in rate codes arises because the benefits of reduced spike count variance cannot compensate for the lower firing rate gain due to adaptation. Since adaptation states have similar dynamics to synaptic responses, the quasi-IID decorrelation transformation of the spike train is plausibly implemented by downstream neurons through matched postsynaptic kinetics. This provides an explanation for observed coding performance in sensory systems that cannot be accounted for by rate coding, for example, at the detection threshold where rate changes can be insignificant.

Understanding how rich dynamics emerge in neural populations requires models exhibiting a wide range of behaviors while remaining interpretable in terms of connectivity and single-neuron dynamics. However, it has been challenging to fit such mechanistic spiking networks at the single-neuron scale to empirical population data. To close this gap, we propose to fit such data at a mesoscale, using a mechanistic but low-dimensional and, hence, statistically tractable model. The mesoscopic representation is obtained by approximating a population of neurons as multiple homogeneous pools of neurons and modeling the dynamics of the aggregate population activity within each pool. We derive the likelihood of both single-neuron and connectivity parameters given this activity, which can then be used to optimize parameters by gradient ascent on the log likelihood or perform Bayesian inference using Markov chain Monte Carlo (MCMC) sampling. We illustrate this approach using a model of generalized integrate-and-fire neurons for which mesoscopic dynamics have been previously derived and show that both single-neuron and connectivity parameters can be recovered from simulated data. In particular, our inference method extracts posterior correlations between model parameters, which define parameter subsets able to reproduce the data. We compute the Bayesian posterior for combinations of parameters using MCMC sampling and investigate how the approximations inherent in a mesoscopic population model affect the accuracy of the inferred single-neuron parameters.

Neural network dynamics are governed by the interaction of spiking neurons. Stochastic aspects of single-neuron dynamics propagate up to the network level and shape the dynamical and informational properties of the population. Mean-field models of population activity disregard the finite-size stochastic fluctuations of network dynamics and thus offer a deterministic description of the system. Here, we derive a stochastic partial differential equation (SPDE) describing the temporal evolution of the finite-size refractory density, which represents the proportion of neurons in a given refractory state at any given time. The population activity-the density of active neurons per unit time-is easily extracted from this refractory density. The SPDE includes finite-size effects through a two-dimensional Gaussian white noise that acts both in time and along the refractory dimension. For an infinite number of neurons the standard mean-field theory is recovered. A discretization of the SPDE along its characteristic curves allows direct simulations of the activity of large but finite spiking networks; this constitutes the main advantage of our approach. Linearizing the SPDE with respect to the deterministic asynchronous state allows the theoretical investigation of finite-size activity fluctuations. In particular, analytical expressions for the power spectrum and autocorrelation of activity fluctuations are obtained. Moreover, our approach can be adapted to incorporate multiple interacting populations and quasi-renewal single-neuron dynamics.

Cancellation of redundant information is a highly desirable feature of sensory systems, since it would potentially lead to a more efficient detection of novel information. However, biologically plausible mechanisms responsible for such selective cancellation, and especially those robust to realistic variations in the intensity of the redundant signals, are mostly unknown. In this work, we study, via in vivo experimental recordings and computational models, the behavior of a cerebellar-like circuit in the weakly electric fish which is known to perform cancellation of redundant stimuli. We experimentally observe contrast invariance in the cancellation of spatially and temporally redundant stimuli in such a system. Our model, which incorporates heterogeneously-delayed feedback, bursting dynamics and burst-induced STDP, is in agreement with our in vivo observations. In addition, the model gives insight on the activity of granule cells and parallel fibers involved in the feedback pathway, and provides a strong prediction on the parallel fiber potentiation time scale. Finally, our model predicts the existence of an optimal learning contrast around 15% contrast levels, which are commonly experienced by interacting fish.

Interactions among animals can result in complex sensory signals containing a variety of socially relevant information, including the number, identity, and relative motion of conspecifics. How the spatiotemporal properties of such evolving naturalistic signals are encoded is a key question in sensory neuroscience. Here, we present results from experiments and modeling that address this issue in the context of the electric sense, which combines the spatial aspects of vision and touch, with the temporal aspects of audition. Wave-type electric fish, such as the brown ghost knifefish, Apteronotus leptorhynchus, used in this study, are uniquely identified by the frequency of their electric organ discharge (EOD). Multiple beat frequencies arise from the superposition of the EODs of each fish. We record the natural electrical signals near the skin of a "receiving" fish that are produced by stationary and freely swimming conspecifics. Using spectral analysis, we find that the primary beats, and the secondary beats between them ("beats of beats"), can be greatly influenced by fish swimming; the resulting motion produces low-frequency envelopes that broaden all the beat peaks and reshape the "noise floor". We assess the consequences of this motion on sensory coding using a model electroreceptor. We show that the primary and secondary beats are encoded in the afferent spike train, but that motion acts to degrade this encoding. We also simulate the response of a realistic population of receptors, and find that it can encode the motion envelope well, primarily due to the receptors with lower firing rates. We discuss the implications of our results for the identification of conspecifics through specific beat frequencies and its possible hindrance by active swimming.

In injured neurons, "leaky" voltage-gated sodium channels (Nav) underlie dysfunctional excitability that ranges from spontaneous subthreshold oscillations (STO), to ectopic (sometimes paroxysmal) excitation, to depolarizing block. In recombinant systems, mechanical injury to Nav1.6-rich membranes causes cytoplasmic Na(+)-loading and "Nav-CLS", i.e., coupled left-(hyperpolarizing)-shift of Nav activation and availability. Metabolic injury of hippocampal neurons (epileptic discharge) results in comparable impairment: left-shifted activation and availability and hence left-shifted I(Na-window). A recent computation study revealed that CLS-based I(Na-window) left-shift dissipates ion gradients and impairs excitability. Here, via dynamical analyses, we focus on sustained excitability patterns in mildly damaged nodes, in particular with more realistic Gaussian-distributed Nav-CLS to mimic "smeared" injury intensity. Since our interest is axons that might survive injury, pumps (sine qua non for live axons) are included. In some simulations, pump efficacy and system volumes are varied. Impacts of current noise inputs are also characterized. The diverse modes of spontaneous rhythmic activity evident in these scenarios are studied using bifurcation analysis. For "mild CLS injury", a prominent feature is slow pump/leak-mediated E(Ion) oscillations. These slow oscillations yield dynamic firing thresholds that underlie complex voltage STO and bursting behaviors. Thus, Nav-CLS, a biophysically justified mode of injury, in parallel with functioning pumps, robustly engenders an emergent slow process that triggers a plethora of pathological excitability patterns. This minimalist "device" could have physiological analogs. At first nodes of Ranvier and at nociceptors, e.g., localized lipid-tuning that modulated Nav midpoints could produce Nav-CLS, as could co-expression of appropriately differing Nav isoforms.

Independent component analysis (ICA) is a powerful method to decouple signals. Most of the algorithms performing ICA do not consider the temporal correlations of the signal, but only higher moments of its amplitude distribution. Moreover, they require some preprocessing of the data (whitening) so as to remove second order correlations. In this paper, we are interested in understanding the neural mechanism responsible for solving ICA. We present an online learning rule that exploits delayed correlations in the input. This rule performs ICA by detecting joint variations in the firing rates of pre- and postsynaptic neurons, similar to a local rate-based Hebbian learning rule.

It is well-known that weakly electric fish can exhibit extreme temporal acuity at the behavioral level, discriminating time intervals in the submicrosecond range. However, relatively little is known about the spatial acuity of the electrosense. Here we use a recently developed model of the electric field generated by Apteronotus leptorhynchus to study spatial acuity and small signal extraction. We show that the quality of sensory information available on the lateral body surface is highest for objects close to the fish's midbody, suggesting that spatial acuity should be highest at this location. Overall, however, this information is relatively blurry and the electrosense exhibits relatively poor acuity. Despite this apparent limitation, weakly electric fish are able to extract the minute signals generated by small prey, even in the presence of large background signals. In fact, we show that the fish's poor spatial acuity may actually enhance prey detection under some conditions. This occurs because the electric image produced by a spatially dense background is relatively "blurred" or spatially uniform. Hence, the small spatially localized prey signal "pops out" when fish motion is simulated. This shows explicitly how the back-and-forth swimming, characteristic of these fish, can be used to generate motion cues that, as in other animals, assist in the extraction of sensory information when signal-to-noise ratios are low. Our study also reveals the importance of the structure of complex electrosensory backgrounds. Whereas large-object spacing is favorable for discriminating the individual elements of a scene, small spacing can increase the fish's ability to resolve a single target object against this background.

In two recent articles, Rudolph and Destexhe (2003, 2005) studied a leaky integrator model (an RC-circuit) driven by correlated ("colored") gaussian conductance noise and Gaussian current noise. In the first article, they derived an expression for the stationary probability density of the membrane voltage; in the second, they modified this expression to cover a larger parameter regime. Here we show by standard analysis of solvable limit cases (white noise limit of additive and multiplicative noise sources; only slow multiplicative noise; only additive noise) and by numerical simulations that their first result does not hold for the general colored-noise case and uncover the errors made in the derivation of a Fokker-Planck equation for the probability density. Furthermore, we demonstrate analytically (including an exact integral expression for the time-dependent mean value of the voltage) and by comparison to simulation results that the extended expression for the probability density works much better but still does not exactly solve the full colored-noise problem. We also show that at stronger synaptic input, the stationary mean value of the linear voltage model may diverge and give an exact condition relating the system parameters for which this takes place.

Oscillatory and synchronized neural activities are commonly found in the brain, and evidence suggests that many of them are caused by global feedback. Their mechanisms and roles in information processing have been discussed often using purely feedforward networks or recurrent networks with constant inputs. On the other hand, real recurrent neural networks are abundant and continually receive information-rich inputs from the outside environment or other parts of the brain. We examine how feedforward networks of spiking neurons with delayed global feedback process information about temporally changing inputs. We show that the network behavior is more synchronous as well as more correlated with and phase-locked to the stimulus when the stimulus frequency is resonant with the inherent frequency of the neuron or that of the network oscillation generated by the feedback architecture. The two eigenmodes have distinct dynamical characteristics, which are supported by numerical simulations and by analytical arguments based on frequency response and bifurcation theory. This distinction is similar to the class I versus class II classification of single neurons according to the bifurcation from quiescence to periodic firing, and the two modes depend differently on system parameters. These two mechanisms may be associated with different types of information processing.

Neuronal adaptation as well as interdischarge interval correlations have been shown to be functionally important properties of physiological neurons. We explore the dynamics of a modified leaky integrate-and-fire (LIF) neuron, referred to as the LIF with threshold fatigue, and show that it reproduces these properties. In this model, the postdischarge threshold reset depends on the preceding sequence of discharge times. We show that in response to various classes of stimuli, namely, constant currents, step currents, white gaussian noise, and sinusoidal currents, the model exhibits new behavior compared with the standard LIF neuron. More precisely, (1) step currents lead to adaptation, that is, a progressive decrease of the discharge rate following the stimulus onset, while in the standard LIF, no such patterns are possible; (2) a saturation in the firing rate occurs in certain regimes, a behavior not seen in the LIF neuron; (3) interspike intervals of the noise-driven modified LIF under constant current are correlated in a way reminiscent of experimental observations, while those of the standard LIF are independent of one another; (4) the magnitude of the correlation coefficients decreases as a function of noise intensity; and (5) the dynamics of the sinusoidally forced modified LIF are described by iterates of an annulus map, an extension to the circle map dynamics displayed by the LIF model. Under certain conditions, this map can give rise to sensitivity to initial conditions and thus chaotic behavior.

We examine the effects of paired delayed excitatory and inhibitory feedback on a single integrate-and-fire neuron with reversal potentials embedded within a feedback network. These effects are studied using bifurcation theory and numerical analysis. The feedback occurs through modulation of the excitatory and inhibitory conductances by the previous firing history of the neuron; as a consequence, the feedback also modifies the membrane time constant. Such paired feedback is ubiquitous in the nervous system. We assume that the feedback dynamics are slower than the membrane time constant, which leads to a rate model formulation. Our article provides an extensive analysis of the possible dynamical behaviors of such simple yet realistic neural loops as a function of the balance between positive and negative feedback, with and without noise, and offers insight into the potential behaviors such loops can exhibit in response to time-varying external inputs. With excitatory feedback, the system can be quiescent, can be periodically firing, or can exhibit bistability between these two states. With inhibitory feedback, quiescence, oscillatory firing rates, and bistability between constant and oscillatory firing-rate solutions are possible. The general case of paired feedback exhibits a blend of the behaviors seen in the extreme cases and can produce chaotic firing. We further derive a condition for a dynamically balanced paired feedback in which there is neither bistability nor oscillations. We also show how a biophysically plausible smoothing of the firing function by noise can modify the existence and stability of fixed points and oscillations of the system. We take advantage in our mathematical analysis of the existence of an invariant manifold, which reduces the dimensionality of the dynamics, and prove the stability of this manifold. The novel computational challenges involved in analyzing such dynamics with and without noise are also described. Our results demonstrate that a paired delayed feedback loop can act as a sophisticated computational unit, capable of switching between a variety of behaviors depending on the input current, the relative strengths and asymmetry of the two parallel feedback pathways, and the delay distributions and noise level.

We study the one-dimensional normal form of a saddle-node system under the influence of additive gaussian white noise and a static "bias current" input parameter, a model that can be looked upon as the simplest version of a type I neuron with stochastic input. This is in contrast with the numerous studies devoted to the noise-driven leaky integrate-and-fire neuron. We focus on the firing rate and coefficient of variation (CV) of the interspike interval density, for which scaling relations with respect to the input parameter and noise intensity are derived. Quadrature formulas for rate and CV are numerically evaluated and compared to numerical simulations of the system and to various approximation formulas obtained in different limiting cases of the model. We also show that caution must be used to extend these results to the Theta neuron model with multiplicative gaussian white noise. The correspondence between the first passage time statistics for the saddle-node model and the Theta neuron model is obtained only in the Stratonovich interpretation of the stochastic Theta neuron model, while previous results have focused only on the Ito interpretation. The correct Stratonovich interpretation yields CVs that are still relatively high, although smaller than in the Ito interpretation; it also produces certain qualitative differences, especially at larger noise intensities. Our analysis provides useful relations for assessing the distance to threshold and the level of synaptic noise in real type I neurons from their firing statistics. We also briefly discuss the effect of finite boundaries (finite values of threshold and reset) on the firing statistics.

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.

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.

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.