Fractional differentiation by neocortical pyramidal neurons

How single neuron properties shape chaotic dynamics and signal transmission in random neural networks

While most models of randomly connected neural networks assume single-neuron models with simple dynamics, neurons in the brain exhibit complex intrinsic dynamics over multiple timescales. We analyze how the dynamical properties of single neurons and recurrent connections interact to shape the effective dynamics in large randomly connected networks. A novel dynamical mean-field theory for strongly connected networks of multi-dimensional rate neurons shows that the power spectrum of the network activity in the chaotic phase emerges from a nonlinear sharpening of the frequency response function of single neurons. For the case of two-dimensional rate neurons with strong adaptation, we find that the network exhibits a state of "resonant chaos", characterized by robust, narrow-band stochastic oscillations. The coherence of stochastic oscillations is maximal at the onset of chaos and their correlation time scales with the adaptation timescale of single units. Surprisingly, the resonance frequency can be predicted from the properties of isolated neurons, even in the presence of heterogeneity in the adaptation parameters. In the presence of these internally-generated chaotic fluctuations, the transmission of weak, low-frequency signals is strongly enhanced by adaptation, whereas signal transmission is not influenced by adaptation in the non-chaotic regime. Our theoretical framework can be applied to other mechanisms at the level of single neurons, such as synaptic filtering, refractoriness or spike synchronization. These results advance our understanding of the interaction between the dynamics of single units and recurrent connectivity, which is a fundamental step toward the description of biologically realistic neural networks.

Distinct Structure of Cortical Population Activity on Fast and Infraslow Timescales

Cortical activity is organized across multiple spatial and temporal scales. Most research on the dynamics of neuronal spiking is concerned with timescales of 1 ms-1 s, and little is known about spiking dynamics on timescales of tens of seconds and minutes. Here, we used frequency domain analyses to study the structure of individual neurons' spiking activity and its coupling to local population rate and to arousal level across 0.01-100 Hz frequency range. In mouse medial prefrontal cortex, the spiking dynamics of individual neurons could be quantitatively captured by a combination of interspike interval and firing rate power spectrum distributions. The relative strength of coherence with local population often differed across timescales: a neuron strongly coupled to population rate on fast timescales could be weakly coupled on slow timescales, and vice versa. On slow but not fast timescales, a substantial proportion of neurons showed firing anticorrelated with the population. Infraslow firing rate changes were largely determined by arousal rather than by local factors, which could explain the timescale dependence of individual neurons' population coupling strength. These observations demonstrate how neurons simultaneously partake in fast local dynamics, and slow brain-wide dynamics, extending our understanding of infraslow cortical activity beyond the mesoscale resolution of fMRI.

Slowing less than 1 Hz is decreased near the seizure onset zone

Focal slowing (<4 Hz) of brain waves is often associated with focal cerebral dysfunction and is assumed to be increased closest to the location of dysfunction. Prior work suggests that slowing may be comprised of at least two distinct neural mechanisms: slow oscillation activity (<1 Hz) may reflect primarily inhibitory cortical mechanisms while power in the delta frequency (1-4 Hz) may correlate with local synaptic strength. In focal epilepsy patients, we examined slow wave activity near and far from the seizure onset zone (SOZ) during wake, sleep, and postictal states using intracranial electroencephalography. We found that slow oscillation (0.3-1 Hz) activity was decreased near the SOZ, while delta activity (2-4 Hz) activity was increased. This finding was most prominent during sleep, and accompanied by a loss of long-range intra-hemispheric synchrony. In contrast to sleep, postictal slowing was characterized by a broadband increase of spectral power, and showed a reduced modulatory effect of slow oscillations on higher frequencies. These results suggest slow oscillation focal slowing is reduced near the seizure onset zone, perhaps reflecting reduced inhibitory activity. Dissociation between slow oscillation and delta slowing could help localize the seizure onset zone from interictal intracranial recordings.

Contrast and luminance adaptation alter neuronal coding and perception of stimulus orientation

Sensory systems face a barrage of stimulation that continually changes along multiple dimensions. These simultaneous changes create a formidable problem for the nervous system, as neurons must dynamically encode each stimulus dimension, despite changes in other dimensions. Here, we measured how neurons in visual cortex encode orientation following changes in luminance and contrast, which are critical for visual processing, but nuisance variables in the context of orientation coding. Using information theoretic analysis and population decoding approaches, we find that orientation discriminability is luminance and contrast dependent, changing over time due to firing rate adaptation. We also show that orientation discrimination in human observers changes during adaptation, in a manner consistent with the neuronal data. Our results suggest that adaptation does not maintain information rates per se, but instead acts to keep sensory systems operating within the limited dynamic range afforded by spiking activity, despite a wide range of possible inputs.

Sensory systems produce stable stimulus representations despite constant changes across multiple stimulus dimensions. Here, the authors reveal dynamic neural coding mechanisms by testing how coding of one dimension (orientation) changes with adaptations to other dimensions (luminance and contrast).

On the chaotic pole of attraction for Hindmarsh-Rose neuron dynamics with external current input

Since the neurologists Hindmarsh and Rose improved the Hodgkin-Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh-Rose neurons with an external current input. Combining with fractional differentiation, the model is generalized with the introduction of an additional parameter, the non-integer order of the derivative σ, and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamics show that in the standard case where the control parameter σ=1, the nerve cell's behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as σ decreases (σ=0.9 and σ=0.85) with the pole of attraction becoming chaotic.

Active dendrites regulate the spatiotemporal spread of signaling microdomains

Microdomains that emerge from spatially constricted spread of biochemical signaling components play a central role in several neuronal computations. Although dendrites, endowed with several voltage-gated ion channels, form a prominent structural substrate for microdomain physiology, it is not known if these channels regulate the spatiotemporal spread of signaling microdomains. Here, we employed a multiscale, morphologically realistic, conductance-based model of the hippocampal pyramidal neuron that accounted for experimental details of electrical and calcium-dependent biochemical signaling. We activated synaptic N-Methyl-d-Aspartate receptors through theta-burst stimulation (TBS) or pairing (TBP) and assessed microdomain propagation along a signaling pathway that included calmodulin, calcium/calmodulin-dependent protein kinase II (CaMKII) and protein phosphatase 1. We found that the spatiotemporal spread of the TBS-evoked microdomain in phosphorylated CaMKII (pCaMKII) was amplified in comparison to that of the corresponding calcium microdomain. Next, we assessed the role of two dendritically expressed inactivating channels, one restorative (A-type potassium) and another regenerative (T-type calcium), by systematically varying their conductances. Whereas A-type potassium channels suppressed the spread of pCaMKII microdomains by altering the voltage response to TBS, T-type calcium channels enhanced this spread by modulating TBS-induced calcium influx without changing the voltage. Finally, we explored cross-dependencies of these channels with other model components, and demonstrated the heavy mutual interdependence of several biophysical and biochemical properties in regulating microdomains and their spread. Our conclusions unveil a pivotal role for dendritic voltage-gated ion channels in actively amplifying or suppressing biochemical signals and their spatiotemporal spread, with critical implications for clustered synaptic plasticity, robust information transfer and efficient neural coding.

The spatiotemporal spread of biochemical signals in neurons and other cells regulate signaling specificity, tuning of signal propagation, along with specificity and clustering of adaptive plasticity. Theoretical and experimental studies have demonstrated a critical role for cellular morphology and the topology of signaling networks in regulating this spread. In this study, we add a significantly complex dimension to this narrative by demonstrating that voltage-gated ion channels on the plasma membrane could actively amplify or suppress the strength and spread of downstream signaling components. Given the expression of different ion channels with wide-ranging heterogeneity in gating kinetics, localization and density, our results point to an increase in complexity of and degeneracy in signaling spread, and unveil a powerful mechanism for regulating biochemical-signaling pathways across different cell types.

Models for Nociception Stimulation and Memory Effects in Awake and Aware Healthy Individuals

OBJECTIVE

This paper introduces a primer in the health care practice, namely a mathematical model and methodology for detecting and analysing nociceptor stimulation followed by related tissue memory effects.

METHODS

Noninvasive nociceptor stimulus protocol and prototype device for measuring bioimpedance is provided. Various time instants, sensor location, and stimulus train have been analysed.

RESULTS

The method and model indicate that nociceptor stimulation perceived as pain in awake healthy volunteers is noninvasively detected. The existence of a memory effect is proven from data. Sensor location had minimal effect on detection level, while day-to-day variability was observed without being significant.

CONCLUSION

Following the experimental study, the model enables a comprehensive management of chronic pain patients, and possibly other analgesia, or pain related regulatory loops.

SIGNIFICANCE

A device and methodology for noninvasive for detecting nociception stimulation have been developed. The proposed method and models have been validated on healthy volunteers.

Behavioral Context Determines Network State and Variability Dynamics in Monkey Motor Cortex

Variability of spiking activity is ubiquitous throughout the brain but little is known about its contextual dependance. Trial-to-trial spike count variability, estimated by the Fano Factor (FF), and within-trial spike time irregularity, quantified by the coefficient of variation (CV), reflect variability on long and short time scales, respectively. We co-analyzed FF and the local coefficient of variation (CV2) in monkey motor cortex comparing two behavioral contexts, movement preparation (wait) and execution (movement). We find that the FF significantly decreases from wait to movement, while the CV2 increases. The more regular firing (expressed by a low CV2) during wait is related to an increased power of local field potential (LFP) beta oscillations and phase locking of spikes to these oscillations. In renewal processes, a widely used model for spiking activity under stationary input conditions, both measures are related as FF ≈ CV². This expectation was met during movement, but not during wait where FF ≫ CV2². Our interpretation is that during movement preparation, ongoing brain processes result in changing network states and thus in high trial-to-trial variability (expressed by a high FF). During movement execution, the network is recruited for performing the stereotyped motor task, resulting in reliable single neuron output. Our interpretation is in the light of recent computational models that generate non-stationary network conditions.

Adaptive coding for dynamic sensory inference

Behavior relies on the ability of sensory systems to infer properties of the environment from incoming stimuli. The accuracy of inference depends on the fidelity with which behaviorally relevant properties of stimuli are encoded in neural responses. High-fidelity encodings can be metabolically costly, but low-fidelity encodings can cause errors in inference. Here, we discuss general principles that underlie the tradeoff between encoding cost and inference error. We then derive adaptive encoding schemes that dynamically navigate this tradeoff. These optimal encodings tend to increase the fidelity of the neural representation following a change in the stimulus distribution, and reduce fidelity for stimuli that originate from a known distribution. We predict dynamical signatures of such encoding schemes and demonstrate how known phenomena, such as burst coding and firing rate adaptation, can be understood as hallmarks of optimal coding for accurate inference.

Descending pathways generate perception of and neural responses to weak sensory input

Natural sensory stimuli frequently consist of a fast time-varying waveform whose amplitude or contrast varies more slowly. While changes in contrast carry behaviorally relevant information necessary for sensory perception, their processing by the brain remains poorly understood to this day. Here, we investigated the mechanisms that enable neural responses to and perception of low-contrast stimuli in the electrosensory system of the weakly electric fish Apteronotus leptorhynchus. We found that fish reliably detected such stimuli via robust behavioral responses. Recordings from peripheral electrosensory neurons revealed stimulus-induced changes in firing activity (i.e., phase locking) but not in their overall firing rate. However, central electrosensory neurons receiving input from the periphery responded robustly via both phase locking and increases in firing rate. Pharmacological inactivation of feedback input onto central electrosensory neurons eliminated increases in firing rate but did not affect phase locking for central electrosensory neurons in response to low-contrast stimuli. As feedback inactivation eliminated behavioral responses to these stimuli as well, our results show that it is changes in central electrosensory neuron firing rate that are relevant for behavior, rather than phase locking. Finally, recordings from neurons projecting directly via feedback to central electrosensory neurons revealed that they provide the necessary input to cause increases in firing rate. Our results thus provide the first experimental evidence that feedback generates both neural and behavioral responses to low-contrast stimuli that are commonly found in the natural environment.

Feedback input from more central to more peripheral brain areas is found ubiquitously in the central nervous system of vertebrates. In this study, we used a combination of electrophysiological, behavioral, and pharmacological approaches to reveal a novel function for feedback pathways in generating neural and behavioral responses to weak sensory input in the weakly electric fish. We first determined that weak sensory input gives rise to responses that are phase locked in both peripheral sensory neurons and in the central neurons that are their downstream targets. However, central neurons also responded to weak sensory inputs that were not relayed via a feedforward input from the periphery, because complete inactivation of the feedback pathway abolished increases in firing rate but not the phase locking in response to weak sensory input. Because such inactivation also abolished the behavioral responses, our results show that the increases in firing rate in central neurons, and not the phase locking, are decoded downstream to give rise to perception. Finally, we discovered that the neurons providing feedback input were also activated by weak sensory input, thereby offering further evidence that feedback is necessary to elicit increases in firing rate that are needed for perception.

Robust Finite-Time Stabilization of Fractional-Order Neural Networks With Discontinuous and Continuous Activation Functions Under Uncertainty

This paper is concerned with robust finite-time stabilization for a class of fractional-order neural networks (FNNs) with two types of activation functions (i.e., discontinuous and continuous activation function) under uncertainty. It is worth noting that there exist few results about FNNs with discontinuous activation functions, which is mainly because classical solutions and theories of differential equations cannot be applied in this case. Especially, there is no relevant finite-time stabilization research for such system, and this paper makes up for the gap. The existence of global solution under the framework of Filippov for such system is guaranteed by limiting discontinuous activation functions. According to set-valued analysis and Kakutani's fixed point theorem, we obtain the existence of equilibrium point. In particular, based on differential inclusion theory and fractional Lyapunov stability theory, several new sufficient conditions are given to ensure finite-time stabilization via a novel discontinuous controller, and the upper bound of the settling time for stabilization is estimated. In addition, we analyze the finite-time stabilization of FNNs with Lipschitz-continuous activation functions under uncertainty. The results of this paper improve corresponding ones of integer-order neural networks with discontinuous and continuous activation functions. Finally, three numerical examples are given to show the effectiveness of the theoretical results.

An integrate-and-fire model to generate spike trains with long-range dependence

Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models. On the other hand, to correctly measure long-range dependence, it is usually necessary to know if the data are stationary. Thus, a methodology to evaluate stationarity of the ISIs is presented and applied to the various IF models. We explain that Markovian IF models may seem to have LRD because of non-stationarities.

Improving Spiking Dynamical Networks: Accurate Delays, Higher-Order Synapses, and Time Cells

Researchers building spiking neural networks face the challenge of improving the biological plausibility of their model networks while maintaining the ability to quantitatively characterize network behavior. In this work, we extend the theory behind the neural engineering framework (NEF), a method of building spiking dynamical networks, to permit the use of a broad class of synapse models while maintaining prescribed dynamics up to a given order. This theory improves our understanding of how low-level synaptic properties alter the accuracy of high-level computations in spiking dynamical networks. For completeness, we provide characterizations for both continuous-time (i.e., analog) and discrete-time (i.e., digital) simulations. We demonstrate the utility of these extensions by mapping an optimal delay line onto various spiking dynamical networks using higher-order models of the synapse. We show that these networks nonlinearly encode rolling windows of input history, using a scale invariant representation, with accuracy depending on the frequency content of the input signal. Finally, we reveal that these methods provide a novel explanation of time cell responses during a delay task, which have been observed throughout hippocampus, striatum, and cortex.

On stability of fixed points and chaos in fractional systems

In this paper, we propose a method to calculate asymptotically period two sinks and define the range of stability of fixed points for a variety of discrete fractional systems of the order 0<α<2. The method is tested on various forms of fractional generalizations of the standard and logistic maps. Based on our analysis, we make a conjecture that chaos is impossible in the corresponding continuous fractional systems.

Fast intensity adaptation enhances the encoding of sound in Drosophila

To faithfully encode complex stimuli, sensory neurons should correct, via adaptation, for stimulus properties that corrupt pattern recognition. Here we investigate sound intensity adaptation in the Drosophila auditory system, which is largely devoted to processing courtship song. Mechanosensory neurons (JONs) in the antenna are sensitive not only to sound-induced antennal vibrations, but also to wind or gravity, which affect the antenna's mean position. Song pattern recognition, therefore, requires adaptation to antennal position (stimulus mean) in addition to sound intensity (stimulus variance). We discover fast variance adaptation in Drosophila JONs, which corrects for background noise over the behaviorally relevant intensity range. We determine where mean and variance adaptation arises and how they interact. A computational model explains our results using a sequence of subtractive and divisive adaptation modules, interleaved by rectification. These results lay the foundation for identifying the molecular and biophysical implementation of adaptation to the statistics of natural sensory stimuli.

Olfactory coding in the turbulent realm

Long-distance olfactory search behaviors depend on odor detection dynamics. Due to turbulence, olfactory signals travel as bursts of variable concentration and spacing and are characterized by long-tail distributions of odor/no-odor events, challenging the computing capacities of olfactory systems. How animals encode complex olfactory scenes to track the plume far from the source remains unclear. Here we focus on the coding of the plume temporal dynamics in moths. We compare responses of olfactory receptor neurons (ORNs) and antennal lobe projection neurons (PNs) to sequences of pheromone stimuli either with white-noise patterns or with realistic turbulent temporal structures simulating a large range of distances (8 to 64 m) from the odor source. For the first time, we analyze what information is extracted by the olfactory system at large distances from the source. Neuronal responses are analyzed using linear-nonlinear models fitted with white-noise stimuli and used for predicting responses to turbulent stimuli. We found that neuronal firing rate is less correlated with the dynamic odor time course when distance to the source increases because of improper coding during long odor and no-odor events that characterize large distances. Rapid adaptation during long puffs does not preclude however the detection of puff transitions in PNs. Individual PNs but not individual ORNs encode the onset and offset of odor puffs for any temporal structure of stimuli. A higher spontaneous firing rate coupled to an inhibition phase at the end of PN responses contributes to this coding property. This allows PNs to decode the temporal structure of the odor plume at any distance to the source, an essential piece of information moths can use in their tracking behavior.

Synchronization of fractional fuzzy cellular neural networks with interactions

In this paper, we introduce fuzzy theory into the fractional cellular neural networks to dynamically enhance the coupling strength and propose a fractional fuzzy neural network model with interactions. Using the Lyapunov principle of fractional differential equations, we design the adaptive control schemes to realize the synchronization and obtain the synchronization criteria. Finally, we provide some numerical examples to show the effectiveness of our obtained results.

LMI Conditions for Global Stability of Fractional-Order Neural Networks

Fractional-order neural networks play a vital role in modeling the information processing of neuronal interactions. It is still an open and necessary topic for fractional-order neural networks to investigate their global stability. This paper proposes some simplified linear matrix inequality (LMI) stability conditions for fractional-order linear and nonlinear systems. Then, the global stability analysis of fractional-order neural networks employs the results from the obtained LMI conditions. In the LMI form, the obtained results include the existence and uniqueness of equilibrium point and its global stability, which simplify and extend some previous work on the stability analysis of the fractional-order neural networks. Moreover, a generalized projective synchronization method between such neural systems is given, along with its corresponding LMI condition. Finally, two numerical examples are provided to illustrate the effectiveness of the established LMI conditions.

Mittag-Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers

In this paper, we propose a fractional-order neural network system with time-varying delays and reaction-diffusion terms. We first develop a new Mittag-Leffler synchronization strategy for the controlled nodes via impulsive controllers. Using the fractional Lyapunov method sufficient conditions are given. We also study the global Mittag-Leffler synchronization of two identical fractional impulsive reaction-diffusion neural networks using linear controllers, which was an open problem even for integer-order models. Since the Mittag-Leffler stability notion is a generalization of the exponential stability concept for fractional-order systems, our results extend and improve the exponential impulsive control theory of neural network system with time-varying delays and reaction-diffusion terms to the fractional-order case. The fractional-order derivatives allow us to model the long-term memory in the neural networks, and thus the present research provides with a conceptually straightforward mathematical representation of rather complex processes. Illustrative examples are presented to show the validity of the obtained results. We show that by means of appropriate impulsive controllers we can realize the stability goal and to control the qualitative behavior of the states. An image encryption scheme is extended using fractional derivatives.

Optimized Parallel Coding of Second-Order Stimulus Features by Heterogeneous Neural Populations

Efficient processing of sensory input is essential to ensure an organism's survival in its natural environment. Growing evidence suggests that sensory neurons can optimally encode natural stimuli by ensuring that their tuning opposes stimulus statistics, such that the resulting neuronal response contains equal power at all frequencies (i.e., is "white"). Such temporal decorrelation or whitening has been observed across modalities, but the effects of neural heterogeneities on determining tuning and thus responses to natural stimuli have not been investigated. Here, we investigate how heterogeneities in sensory pyramidal neurons organized in three parallel maps representing the body surface determine responses to second-order electrosensory stimulus features in the weakly electric fish Apteronotus leptorhynchus While some sources of heterogeneities such as ON- and OFF-type responses to first-order did not affect responses to second-order electrosensory stimulus features, other sources of heterogeneity within and across the maps strongly determined responses. We found that these cells effectively performed a fractional differentiation operation on their input with exponents ranging from zero (no differentiation) to 0.4 (strong differentiation). Varying adaptation in a simple model explained these heterogeneities and predicted a strong correlation between fractional differentiation and adaptation. Using natural stimuli, we found that only a small fraction of neurons implemented temporal whitening. Rather, a large fraction of neurons did not perform any significant whitening and thus preserved natural input statistics in their responses. We propose that this information is needed to properly decode optimized information sent in parallel through temporally whitened responses based on context.

SIGNIFICANCE STATEMENT

We demonstrate that heterogeneities in the same sensory neuron type can either have no or significant influence on their responses to second-order stimulus features. While an ON- or OFF-type response to first-order stimulus attributes has no significant influence on responses to second-order stimulus features, we found that only a small fraction of sensory neurons optimally encoded natural stimuli through high-pass filtering, thereby implementing temporal whitening. Surprisingly, a large fraction of sensory neurons performed little if no filtering of stimuli, thereby preserving natural stimulus statistics. We hypothesize that this pathway is necessary to properly decode optimized information contained in temporally whitened responses based on context.

Olfactory receptor neurons use gain control and complementary kinetics to encode intermittent odorant stimuli

Insects find food and mates by navigating odorant plumes that can be highly intermittent, with intensities and durations that vary rapidly over orders of magnitude. Much is known about olfactory responses to pulses and steps, but it remains unclear how olfactory receptor neurons (ORNs) detect the intensity and timing of natural stimuli, where the absence of scale in the signal makes detection a formidable olfactory task. By stimulating Drosophila ORNs in vivo with naturalistic and Gaussian stimuli, we show that ORNs adapt to stimulus mean and variance, and that adaptation and saturation contribute to naturalistic sensing. Mean-dependent gain control followed the Weber-Fechner relation and occurred primarily at odor transduction, while variance-dependent gain control occurred at both transduction and spiking. Transduction and spike generation possessed complementary kinetic properties, that together preserved the timing of odorant encounters in ORN spiking, regardless of intensity. Such scale-invariance could be critical during odor plume navigation.

Envelope statistics of self-motion signals experienced by human subjects during everyday activities: Implications for vestibular processing

There is accumulating evidence that the brain's neural coding strategies are constrained by natural stimulus statistics. Here we investigated the statistics of the time varying envelope (i.e. a second-order stimulus attribute that is related to variance) of rotational and translational self-motion signals experienced by human subjects during everyday activities. We found that envelopes can reach large values across all six motion dimensions (~450 deg/s for rotations and ~4 G for translations). Unlike results obtained in other sensory modalities, the spectral power of envelope signals decreased slowly for low (< 2 Hz) and more sharply for high (>2 Hz) temporal frequencies and thus was not well-fit by a power law. We next compared the spectral properties of envelope signals resulting from active and passive self-motion, as well as those resulting from signals obtained when the subject is absent (i.e. external stimuli). Our data suggest that different mechanisms underlie deviation from scale invariance in rotational and translational self-motion envelopes. Specifically, active self-motion and filtering by the human body cause deviation from scale invariance primarily for translational and rotational envelope signals, respectively. Finally, we used well-established models in order to predict the responses of peripheral vestibular afferents to natural envelope stimuli. We found that irregular afferents responded more strongly to envelopes than their regular counterparts. Our findings have important consequences for understanding the coding strategies used by the vestibular system to process natural second-order self-motion signals.

Fractional-order leaky integrate-and-fire model with long-term memory and power law dynamics

Pyramidal neurons produce different spiking patterns to process information, communicate with each other and transform information. These spiking patterns have complex and multiple time scale dynamics that have been described with the fractional-order leaky integrate-and-Fire (FLIF) model. Models with fractional (non-integer) order differentiation that generalize power law dynamics can be used to describe complex temporal voltage dynamics. The main characteristic of FLIF model is that it depends on all past values of the voltage that causes long-term memory. The model produces spikes with high interspike interval variability and displays several spiking properties such as upward spike-frequency adaptation and long spike latency in response to a constant stimulus. We show that the subthreshold voltage and the firing rate of the fractional-order model make transitions from exponential to power law dynamics when the fractional order α decreases from 1 to smaller values. The firing rate displays different types of spike timing adaptation caused by changes on initial values. We also show that the voltage-memory trace and fractional coefficient are the causes of these different types of spiking properties. The voltage-memory trace that represents the long-term memory has a feedback regulatory mechanism and affects spiking activity. The results suggest that fractional-order models might be appropriate for understanding multiple time scale neuronal dynamics. Overall, a neuron with fractional dynamics displays history dependent activities that might be very useful and powerful for effective information processing.

Evolution of moments and correlations in nonrenewal escape-time processes

The theoretical description of nonrenewal stochastic systems is a challenge. Analytical results are often not available or can be obtained only under strong conditions, limiting their applicability. Also, numerical results have mostly been obtained by ad hoc Monte Carlo simulations, which are usually computationally expensive when a high degree of accuracy is needed. To gain quantitative insight into these systems under general conditions, we here introduce a numerical iterated first-passage time approach based on solving the time-dependent Fokker-Planck equation (FPE) to describe the statistics of nonrenewal stochastic systems. We illustrate the approach using spike-triggered neuronal adaptation in the leaky and perfect integrate-and-fire model, respectively. The transition to stationarity of first-passage time moments and their sequential correlations occur on a nontrivial time scale that depends on all system parameters. Surprisingly this is so for both single exponential and scale-free power-law adaptation. The method works beyond the small noise and time-scale separation approximations. It shows excellent agreement with direct Monte Carlo simulations, which allow for the computation of transient and stationary distributions. We compare different methods to compute the evolution of the moments and serial correlation coefficients (SCCs) and discuss the challenge of reliably computing the SCCs, which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate-and-fire models. In conclusion, our methods provide a general picture of nonrenewal dynamics in a wide range of stochastic systems exhibiting short- and long-range correlations.

Dynamical Timescale Explains Marginal Stability in Excitability Dynamics

Action potentials, taking place over milliseconds, are the basis of neural computation. However, the dynamics of excitability over longer, behaviorally relevant timescales remain underexplored. A recent experiment used long-term recordings from single neurons to reveal multiple timescale fluctuations in response to constant stimuli, along with more reliable responses to variable stimuli. Here, we demonstrate that this apparent paradox is resolved if neurons operate in a marginally stable dynamic regime, which we reveal using a novel inference method. Excitability in this regime is characterized by large fluctuations while retaining high sensitivity to external varying stimuli. A new model with a dynamic recovery timescale that interacts with excitability captures this dynamic regime and predicts the neurons' response with high accuracy. The model explains most experimental observations under several stimulus statistics. The compact structure of our model permits further exploration on the network level.SIGNIFICANCE STATEMENT Excitability is the basis for all neural computations and its long-term dynamics reveal a complex combination of many timescales. We discovered that neural excitability operates under a marginally stable regime in which the system is dominated by internal fluctuation while retaining high sensitivity to externally varying stimuli. We offer a novel approach to modeling excitability dynamics by assuming that the recovery timescale is itself a dynamic variable. Our model is able to capture a wide range of experimental phenomena using few parameters with significantly higher predictive power than previous models.

SK channel subtypes enable parallel optimized coding of behaviorally relevant stimulus attributes: A review

Ion channels play essential roles toward determining how neurons respond to sensory input to mediate perception and behavior. Small conductance calcium-activated potassium (SK) channels are found ubiquitously throughout the brain and have been extensively characterized both molecularly and physiologically in terms of structure and function. It is clear that SK channels are key determinants of neural excitability as they mediate important neuronal response properties such as spike frequency adaptation. However, the functional roles of the different known SK channel subtypes are not well understood. Here we review recent evidence from the electrosensory system of weakly electric fish suggesting that the function of different SK channel subtypes is to optimize the processing of independent but behaviorally relevant stimulus attributes. Indeed, natural sensory stimuli frequently consist of a fast time-varying waveform (i.e., the carrier) whose amplitude (i.e., the envelope) varies slowly and independently. We first review evidence showing how somatic SK2 channels mediate tuning and responses to carrier waveforms. We then review evidence showing how dendritic SK1 channels instead determine tuning and optimize responses to envelope waveforms based on their statistics as found in the organism's natural environment in an independent fashion. The high degree of functional homology between SK channels in electric fish and their mammalian orthologs, as well as the many important parallels between the electrosensory system and the mammalian visual, auditory, and vestibular systems, suggest that these functional roles are conserved across systems and species.

Electrosensory processing in Apteronotus albifrons: implications for general and specific neural coding strategies across wave-type weakly electric fish species

Understanding how the brain processes sensory input to generate behavior remains an important problem in neuroscience. Towards this end, it is useful to compare results obtained across multiple species to gain understanding as to the general principles of neural coding. Here we investigated hindbrain pyramidal cell activity in the weakly electric fish Apteronotus albifrons We found strong heterogeneities when looking at baseline activity. Additionally, ON- and OFF-type cells responded to increases and decreases of sinusoidal and noise stimuli, respectively. While both cell types displayed band-pass tuning, OFF-type cells were more broadly tuned than their ON-type counterparts. The observed heterogeneities in baseline activity as well as the greater broadband tuning of OFF-type cells were both similar to those previously reported in other weakly electric fish species, suggesting that they constitute general features of sensory processing. However, we found that peak tuning occurred at frequencies ∼15 Hz in A. albifrons, which is much lower than values reported in the closely related species Apteronotus leptorhynchus and the more distantly related species Eigenmannia virescens In response to stimuli with time-varying amplitude (i.e., envelope), ON- and OFF-type cells displayed similar high-pass tuning curves characteristic of fractional differentiation and possibly indicate optimized coding. These tuning curves were qualitatively similar to those of pyramidal cells in the closely related species A. leptorhynchus In conclusion, comparison between our and previous results reveals general and species-specific neural coding strategies. We hypothesize that differences in coding strategies, when observed, result from different stimulus distributions in the natural/social environment.

Rapid Sensory Adaptation Redux: A Circuit Perspective

Adaptation is fundamental to life. All organisms adapt over timescales that span from evolution to generations and lifetimes to moment-by-moment interactions. The nervous system is particularly adept at rapidly adapting to change, and this in fact may be one of its fundamental principles of organization and function. Rapid forms of sensory adaptation have been well documented across all sensory modalities in a wide range of organisms, yet we do not have a comprehensive understanding of the adaptive cellular mechanisms that ultimately give rise to the corresponding percepts, due in part to the complexity of the circuitry. In this Perspective, we aim to build links between adaptation at multiple scales of neural circuitry by investigating the differential adaptation across brain regions and sub-regions and across specific cell types, for which the explosion of modern tools has just begun to enable. This investigation points to a set of challenges for the field to link functional observations to adaptive properties of the neural circuit that ultimately underlie percepts.

Efficient sensory cortical coding optimizes pursuit eye movements

In the natural world, the statistics of sensory stimuli fluctuate across a wide range. In theory, the brain could maximize information recovery if sensory neurons adaptively rescale their sensitivity to the current range of inputs. Such adaptive coding has been observed in a variety of systems, but the premise that adaptation optimizes behaviour has not been tested. Here we show that adaptation in cortical sensory neurons maximizes information about visual motion in pursuit eye movements guided by that cortical activity. We find that gain adaptation drives a rapid (<100 ms) recovery of information after shifts in motion variance, because the neurons and behaviour rescale their sensitivity to motion fluctuations. Both neurons and pursuit rapidly adopt a response gain that maximizes motion information and minimizes tracking errors. Thus, efficient sensory coding is not simply an ideal standard but a description of real sensory computation that manifests in improved behavioural performance.

Impulsive synchronization of fractional Takagi-Sugeno fuzzy complex networks

This paper focuses on impulsive synchronization of fractional Takagi-Sugeno (T-S) fuzzy complex networks. A novel comparison principle is built for the fractional impulsive system. Then a synchronization criterion is established for the fractional T-S fuzzy complex networks by utilizing the comparison principle. The method is also illustrated by applying the fractional T-S fuzzy Rössler's complex networks.