Fractional differentiation by neocortical pyramidal neurons

Intermittent spike train processing through fractional leaky integrate-and-fire neuromorphic unit

The leaky integrate-and-fire (LIF) model provides a fundamental framework for modeling neuronal dynamics in spiking networks. While generalized LIF models can incorporate features, such as spike-frequency adaptation and noise, our study specifically examines its fractional-order extension governed by a relaxation equation with a fractional derivative, whose power-law dynamics emulate long-term memory effects ideal for processing intermittent, scale-invariant signals. Statistical properties of the response of the fractional-order LIF model to a flickering input voltage pulse flow, characterized by a fractional Poisson process of order ν, are evaluated. To implement the fractional LIF model in hardware, we developed a microscale transistor using a network of single-walled carbon nanotubes with an electrolyte gate. The system exhibits fractional-order dynamics, making it well-suited for neuromorphic spiking networks that process scale-invariant signals with long-range temporal correlations.

A prospective code for value in the serotonin system

The in vivo responses of dorsal raphe nucleus serotonin neurons to emotionally salient stimuli are a puzzle1. Existing theories centring on reward2, surprise3, salience4 and uncertainty5 individually account for some aspects of serotonergic activity but not others. Merging ideas from reinforcement learning theory6 with recent insights into the filtering properties of the dorsal raphe nucleus7, here we find a unifying perspective in a prospective code for value. This biological code for near-future reward explains why serotonin neurons are activated by both rewards and punishments3,4,8-13, and why these neurons are more strongly activated by surprising rewards but have no such surprise preference for punishments3,9-observations that previous theories have failed to reconcile. Finally, our model quantitatively predicts in vivo population activity better than previous theories. By reconciling previous theories and establishing a precise connection with reinforcement learning, our work represents an important step towards understanding the role of serotonin in learning and behaviour.

Power-law adaptation in the presynaptic vesicle cycle

After synaptic transmission, fused synaptic vesicles are recycled, enabling the synapse to recover its capacity for renewed release. The recovery steps, which range from endocytosis to vesicle docking and priming, have been studied individually, but it is not clear what their impact on the overall dynamics of synaptic recycling is, and how they influence signal transmission. Here we model the dynamics of vesicle recycling and find that the multiple timescales of the recycling steps are reflected in synaptic recovery. This leads to multi-timescale synapse dynamics, which can be described by a simplified synaptic model with 'power-law' adaptation. Using cultured hippocampal neurons, we test this model experimentally, and show that the duration of synaptic exhaustion changes the effective synaptic recovery timescale, as predicted by the model. Finally, we show that this adaptation could implement a specific function in the hippocampus, namely enabling efficient communication between neurons through the temporal whitening of hippocampal spike trains.

Neural networks with optimized single-neuron adaptation uncover biologically plausible regularization

Neurons in the brain have rich and adaptive input-output properties. Features such as heterogeneous f-I curves and spike frequency adaptation are known to place single neurons in optimal coding regimes when facing changing stimuli. Yet, it is still unclear how brain circuits exploit single-neuron flexibility, and how network-level requirements may have shaped such cellular function. To answer this question, a multi-scaled approach is needed where the computations of single neurons and neural circuits must be considered as a complete system. In this work, we use artificial neural networks to systematically investigate single-neuron input-output adaptive mechanisms, optimized in an end-to-end fashion. Throughout the optimization process, each neuron has the liberty to modify its nonlinear activation function parametrized to mimic f-I curves of biological neurons, either by learning an individual static function or via a learned and shared adaptation mechanism to modify activation functions in real-time during a task. We find that such adaptive networks show much-improved robustness to noise and changes in input statistics. Using tools from dynamical systems theory, we analyze the role of these emergent single-neuron properties and argue that neural diversity and adaptation play an active regularization role, enabling neural circuits to optimally propagate information across time. Finally, we outline similarities between these optimized solutions and known coding strategies found in biological neurons, such as gain scaling and fractional order differentiation/integration.

Stability and synchronization of fractional-order reaction-diffusion inertial time-delayed neural networks with parameters perturbation

This study is centered around the dynamic behaviors observed in a class of fractional-order generalized reaction-diffusion inertial neural networks (FGRDINNs) with time delays. These networks are characterized by differential equations involving two distinct fractional derivatives of the state. The global uniform stability of FGRDINNs with time delays is explored utilizing Lyapunov comparison principles. Furthermore, global synchronization conditions for FGRDINNs with time delays are derived through the Lyapunov direct method, with consideration given to various feedback control strategies and parameter perturbations. The effectiveness of the theoretical findings is demonstrated through three numerical examples, and the impact of controller parameters on the error system is further investigated.

A Biomimetic Visual Detection Model: Event-Driven LGMDs Implemented With Fractional Spiking Neuron Circuits

OBJECTIVE

Lobula giant motion detectors (LGMDs) in locusts effectively predict collisions and trigger avoidance, with potential applications in autonomous driving and UAVs. Research on LGMD characteristics splits into two views: one focusing on the presynaptic visual pathway, the other on the postsynaptic LGMD neurons. Both perspectives have support, leading to two computational models, but they lack a biophysical description of the individual LGMD neuron behavior. This paper aims to mimic and explain LGMD behavior based on fractional spiking neurons (FSNs) and construct a biomimetic visual model for the LGMD compatible with these characteristics.

METHODS

We implement the visual model using an event camera to simulate photoreceptors and follow the ON/OFF visual pathway, incorporating lateral inhibition to mimic the LGMD system from the bottom up. Second, most computational models of motion perception use only the dendrites within the LGMD neurons as the ideal pathway for linear summation, ignoring dendritic effects inducing neuronal properties. Thus, we introduce FSN circuits by altering dendritic morphological parameters to simulate multi-scale spike frequency adaptation (SFA) observed in LGMDs. Additionally, we add one more circuit of dendritic trees into the FSNs to be compatible with the postsynaptic feed-forward inhibition (FFI) in LGMD neurons, providing a novel explanatory and predictive model.

RESULTS

We test that the event-driven biomimetic visual model can achieve collision detection and looming selection in different complex scenes, especially fast-moving objects.

Signatures of hierarchical temporal processing in the mouse visual system

A core challenge for the brain is to process information across various timescales. This could be achieved by a hierarchical organization of temporal processing through intrinsic mechanisms (e.g., recurrent coupling or adaptation), but recent evidence from spike recordings of the rodent visual system seems to conflict with this hypothesis. Here, we used an optimized information-theoretic and classical autocorrelation analysis to show that information- and correlation timescales of spiking activity increase along the anatomical hierarchy of the mouse visual system under visual stimulation, while information-theoretic predictability decreases. Moreover, intrinsic timescales for spontaneous activity displayed a similar hierarchy, whereas the hierarchy of predictability was stimulus-dependent. We could reproduce these observations in a basic recurrent network model with correlated sensory input. Our findings suggest that the rodent visual system employs intrinsic mechanisms to achieve longer integration for higher cortical areas, while simultaneously reducing predictability for an efficient neural code.

Effects of high-order interactions on synchronization of a fractional-order neural system

In this study, effects of high-order interactions on synchronization of the fractional-order Hindmarsh-Rose neuron models have been examined deeply. Three different network situations in which first-order coupling, high-order couplings and first-plus second-order couplings included in the neuron models, have been considered, respectively. In order to find the optimal values of the first- and high-order coupling parameters by minimizing the cost function resulted from pairwise and triple interactions, the particle swarm optimization algorithm is employed. It has been deduced from the numerical simulation results that the first-plus second-order couplings induce the synchronization with both reduced first-order coupling strength and total cost compared to the first-order coupled case solely. When the only first-order coupled case is compared with the only second-order coupled case, it is determined that the neural network with only second-order couplings involved could achieve synchronization with lower coupling strength and, as a natural result, lower cost. On the other hand, solely second- and first-plus second-order coupled networks give very similar results each other. Therefore, high-order interactions have a positive effect on the synchronization. Additionally, increasing the network size decreases the values of the both first- and high-order coupling strengths to reach synchronization. However, in this case, total cost should be kept in the mind. Decreasing the fractional order parameter causes slower synchronization due to the decreased frequency of the neural response. On the other hand, more synchronous network is possible with increasing the fractional order parameter. Thus, the neural network with higher fractional order as well as high-order coupled is a good candidate in terms of the neural synchronization.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-023-10055-z.

Spiking attractor model of motor cortex explains modulation of neural and behavioral variability by prior target information

When preparing a movement, we often rely on partial or incomplete information, which can decrement task performance. In behaving monkeys we show that the degree of cued target information is reflected in both, neural variability in motor cortex and behavioral reaction times. We study the underlying mechanisms in a spiking motor-cortical attractor model. By introducing a biologically realistic network topology where excitatory neuron clusters are locally balanced with inhibitory neuron clusters we robustly achieve metastable network activity across a wide range of network parameters. In application to the monkey task, the model performs target-specific action selection and accurately reproduces the task-epoch dependent reduction of trial-to-trial variability in vivo where the degree of reduction directly reflects the amount of processed target information, while spiking irregularity remained constant throughout the task. In the context of incomplete cue information, the increased target selection time of the model can explain increased behavioral reaction times. We conclude that context-dependent neural and behavioral variability is a signum of attractor computation in the motor cortex.

The tuning of tuning: How adaptation influences single cell information transfer

Sensory neurons reconstruct the world from action potentials (spikes) impinging on them. To effectively transfer information about the stimulus to the next processing level, a neuron needs to be able to adapt its working range to the properties of the stimulus. Here, we focus on the intrinsic neural properties that influence information transfer in cortical neurons and how tightly their properties need to be tuned to the stimulus statistics for them to be effective. We start by measuring the intrinsic information encoding properties of putative excitatory and inhibitory neurons in L2/3 of the mouse barrel cortex. Excitatory neurons show high thresholds and strong adaptation, making them fire sparsely and resulting in a strong compression of information, whereas inhibitory neurons that favour fast spiking transfer more information. Next, we turn to computational modelling and ask how two properties influence information transfer: 1) spike-frequency adaptation and 2) the shape of the IV-curve. We find that a subthreshold (but not threshold) adaptation, the 'h-current', and a properly tuned leak conductance can increase the information transfer of a neuron, whereas threshold adaptation can increase its working range. Finally, we verify the effect of the IV-curve slope in our experimental recordings and show that excitatory neurons form a more heterogeneous population than inhibitory neurons. These relationships between intrinsic neural features and neural coding that had not been quantified before will aid computational, theoretical and systems neuroscientists in understanding how neuronal populations can alter their coding properties, such as through the impact of neuromodulators. Why the variability of intrinsic properties of excitatory neurons is larger than that of inhibitory ones is an exciting question, for which future research is needed.

Asymptotic stability of nonlinear fractional delay differential equations with α ∈ (1, 2): An application to fractional delay neural networks

We introduce a theorem on linearized asymptotic stability for nonlinear fractional delay differential equations (FDDEs) with a Caputo order α∈(1,2), which can be directly used for fractional delay neural networks. It relies on three technical tools: a detailed root analysis for the characteristic equation, estimation for the generalized Mittag-Leffler function, and Lyapunov's first method. We propose coefficient-type criteria to ensure the stability of linear FDDEs through a detailed root analysis for the characteristic equation obtained by the Laplace transform. Further, under the criteria, we provide a wise expression of the generalized Mittag-Leffler functions and prove their polynomial long-time decay rates. Utilizing the well-established Lyapunov's first method, we establish that an equilibrium of a nonlinear Caputo FDDE attains asymptotically stability if its linearization system around the equilibrium solution is asymptotically stable. Finally, as a by-product of our results, we explicitly describe the asymptotic properties of fractional delay neural networks. To illustrate the effectiveness of our theoretical results, numerical simulations are also presented.

Fractional order memcapacitive neuromorphic elements reproduce and predict neuronal function

There is an increasing need to implement neuromorphic systems that are both energetically and computationally efficient. There is also great interest in using electric elements with memory, memelements, that can implement complex neuronal functions intrinsically. A feature not widely incorporated in neuromorphic systems is history-dependent action potential time adaptation which is widely seen in real cells. Previous theoretical work shows that power-law history dependent spike time adaptation, seen in several brain areas and species, can be modeled with fractional order differential equations. Here, we show that fractional order spiking neurons can be implemented using super-capacitors. The super-capacitors have fractional order derivative and memcapacitive properties. We implemented two circuits, a leaky integrate and fire and a Hodgkin-Huxley. Both circuits show power-law spiking time adaptation and optimal coding properties. The spiking dynamics reproduced previously published computer simulations. However, the fractional order Hodgkin-Huxley circuit showed novel dynamics consistent with criticality. We compared the responses of this circuit to recordings from neurons in the weakly-electric fish that have previously been shown to perform fractional order differentiation of their sensory input. The criticality seen in the circuit was confirmed in spontaneous recordings in the live fish. Furthermore, the circuit also predicted long-lasting stimulation that was also corroborated experimentally. Our work shows that fractional order memcapacitors provide intrinsic memory dependence that could allow implementation of computationally efficient neuromorphic devices. Memcapacitors are static elements that consume less energy than the most widely studied memristors, thus allowing the realization of energetically efficient neuromorphic devices.

Dynamics and synchronization control of fractional conformable neuron system

Dynamic analysis, electrical coupling and synchronization control of the conformable FitzHugh-Nagumo neuronal models have been presented in this work. Firstly, equations of the Adomian-Decomposition-Method and conformable neuron model have been introduced. The Adomian-Decomposition-Method has been employed for the numerical simulation analysis, since it converges fast and provides serial solutions. Fractional order and external current stimulus have been considered as bifurcation parameters and their effects on neuron model dynamics have been examined in detail. Then, the electrical coupling of the two conformable neuronal models without any controller has been revealed and the significance of the coupling strength parameter has been evaluated. To eliminate impact of the coupling strength parameter on synchronization status of neurons, Lyapunov control method has been employed for synchronization control. In the last step, the numerical simulation studies have been experimentally verified using the Texas Instrument Delfino digital signal processor board. Numerical simulation results together with experimental validation have showed that the types of dynamics of the related neuron model are not affected from the change of the fractional order of conformable derivative, but the frequency of the dynamic response of the neuronal model is changed from the alteration of the fractional order. The frequency of response of the neuron model increases with decreasing values of the fractional order. On the other hand, if there is no synchronization control method, the coupled neuron models exhibit response ranging from synchronous to asynchronous depending on the sign and value of the coupling parameter. Additionally, decreasing values of the fractional order may allow the coupled neurons to enter the synchronous state more quickly due to increasing frequency of response of the neuronal model. Finally, the coupled neuron models could exhibit synchronous behavior, that is determined by calculating the standard deviation results, regardless of the value of the coupling parameter by using the Lyapunov control method.

Complex adaptive learning cortical neural network systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviours

Complex networks have been programmed to mimic the input and output functions in multiple biophysical algorithms of cortical neurons at spiking resolution. Prior research has demonstrated that the ineffectual features of membranes can be taken into account by discrete fractional commensurate, non-commensurate and variable-order patterns, which may generate multiple kinds of memory-dependent behaviour. Firing structures involving regular resonator chattering, fast, chaotic spiking and chaotic bursts play important roles in cortical nerve cell insights and execution. Yet, it is unclear how extensively the behaviour of discrete fractional-order excited mechanisms can modify firing cell attributes. It is illustrated that the discrete fractional behaviour of the Izhikevich neuron framework can generate an assortment of resonances for cortical activity via the aforesaid scheme. We analyze the bifurcation using fragmenting periodic solutions to demonstrate the evolution of periods in the framework's behaviour. We investigate various bursting trends both conceptually and computationally with the fractional difference equation. Additionally, the consequences of an excitable and inhibited Izhikevich neuron network (INN) utilizing a regulated factor set exhibit distinctive dynamic actions depending on fractional exponents regulating over extended exchanges. Ultimately, dynamic controllers for stabilizing and synchronizing the suggested framework are shown. This special spiking activity and other properties of the fractional-order model are caused by the memory trace that emerges from the fractional-order dynamics and integrates all the past activities of the neuron. Our results suggest that the complex dynamics of spiking and bursting can be the result of the long-term dependence and interaction of intracellular and extracellular ionic currents.

The role of long-term power-law memory in controlling large-scale dynamical networks

Controlling large-scale dynamical networks is crucial to understand and, ultimately, craft the evolution of complex behavior. While broadly speaking we understand how to control Markov dynamical networks, where the current state is only a function of its previous state, we lack a general understanding of how to control dynamical networks whose current state depends on states in the distant past (i.e. long-term memory). Therefore, we require a different way to analyze and control the more prevalent long-term memory dynamical networks. Herein, we propose a new approach to control dynamical networks exhibiting long-term power-law memory dependencies. Our newly proposed method enables us to find the minimum number of driven nodes (i.e. the state vertices in the network that are connected to one and only one input) and their placement to control a long-term power-law memory dynamical network given a specific time-horizon, which we define as the 'time-to-control'. Remarkably, we provide evidence that long-term power-law memory dynamical networks require considerably fewer driven nodes to steer the network's state to a desired goal for any given time-to-control as compared with Markov dynamical networks. Finally, our method can be used as a tool to determine the existence of long-term memory dynamics in networks.

An advanced approach for the electrical responses of discrete fractional-order biophysical neural network models and their dynamical responses

The multiple activities of neurons frequently generate several spiking-bursting variations observed within the neurological mechanism. We show that a discrete fractional-order activated nerve cell framework incorporating a Caputo-type fractional difference operator can be used to investigate the impacts of complex interactions on the surge-empowering capabilities noticed within our findings. The relevance of this expansion is based on the model's structure as well as the commensurate and incommensurate fractional-orders, which take kernel and inherited characteristics into account. We begin by providing data regarding the fluctuations in electronic operations using the fractional exponent. We investigate two-dimensional Morris-Lecar neuronal cell frameworks via spiked and saturated attributes, as well as mixed-mode oscillations and mixed-mode bursting oscillations of a decoupled fractional-order neuronal cell. The investigation proceeds by using a three-dimensional slow-fast Morris-Lecar simulation within the fractional context. The proposed method determines a method for describing multiple parallels within fractional and integer-order behaviour. We examine distinctive attribute environments where inactive status develops in detached neural networks using stability and bifurcation assessment. We demonstrate features that are in accordance with the analysis's findings. The Erdös-Rényi connection of asynchronization transformed neural networks (periodic and actionable) is subsequently assembled and paired via membranes that are under pressure. It is capable of generating multifaceted launching processes in which dormant neural networks begin to come under scrutiny. Additionally, we demonstrated that boosting connections can cause classification synchronization, allowing network devices to activate in conjunction in the future. We construct a reduced-order simulation constructed around clustering synchronisation that may represent the operations that comprise the whole system. Our findings indicate the influence of fractional-order is dependent on connections between neurons and the system's stored evidence. Moreover, the processes capture the consequences of fractional derivatives on surge regularity modification and enhance delays that happen across numerous time frames in neural processing.

A biophysical perspective on the resilience of neuronal excitability across timescales

Neuronal membrane excitability must be resilient to perturbations that can take place over timescales from milliseconds to months (or even years in long-lived animals). A great deal of attention has been paid to classes of homeostatic mechanisms that contribute to long-term maintenance of neuronal excitability through processes that alter a key structural parameter: the number of ion channel proteins present at the neuronal membrane. However, less attention has been paid to the self-regulating 'automatic' mechanisms that contribute to neuronal resilience by virtue of the kinetic properties of ion channels themselves. Here, we propose that these two sets of mechanisms are complementary instantiations of feedback control, together enabling resilience on a wide range of temporal scales. We further point to several methodological and conceptual challenges entailed in studying these processes - both of which involve enmeshed feedback control loops - and consider the consequences of these mechanisms of resilience.

Multiple Mittag-Leffler Stability of Fractional-Order Complex-Valued Memristive Neural Networks With Delays

This article discusses the coexistence and dynamical behaviors of multiple equilibrium points (Eps) for fractional-order complex-valued memristive neural networks (FCVMNNs) with delays. First, based on the state space partition method, some sufficient conditions are proposed to guarantee that there are multiple Eps in one FCVMNN. Then, the Mittag-Leffler stability of those multiple Eps is proved by using the Lyapunov function. Simultaneously, the enlarged attraction basins are obtained to improve and extend the existing theoretical results in the previous literature. In addition, some existing stability results in the literature are special cases of a new result herein. Finally, two illustrative examples with computer simulations are presented to verify the effectiveness of theoretical analysis.

Logarithmically scaled, gamma distributed neuronal spiking

Naturally log-scaled quantities abound in the nervous system. Distributions of these quantities have non-intuitive properties, which have implications for data analysis and the understanding of neural circuits. Here, we review the log-scaled statistics of neuronal spiking and the relevant analytical probability distributions. Recent work using log-scaling revealed that interspike intervals of forebrain neurons segregate into discrete modes reflecting spiking at different timescales and are each well-approximated by a gamma distribution. Each neuron spends most of the time in an irregular spiking 'ground state' with the longest intervals, which determines the mean firing rate of the neuron. Across the entire neuronal population, firing rates are log-scaled and well approximated by the gamma distribution, with a small number of highly active neurons and an overabundance of low rate neurons (the 'dark matter'). These results are intricately linked to a heterogeneous balanced operating regime, which confers upon neuronal circuits multiple computational advantages and has evolutionarily ancient origins.

Power mapping-based stability analysis and order adjustment control for fractional-order multiple delayed systems

This article investigates the asymptotic stability of a general class of fractional-order multiple delayed systems to evaluate the delay robustness. We establish a one-to-one spectral connection between the original fractional-order system and the transformed one under the power mapping. The applicability of the Cluster Treatment of Characteristic Roots paradigm to the transformed dynamics is proved by this connection. Then, we utilize the Dixon resultant-based frequency sweeping framework to create the complete stability map. The results demonstrate that the order adjustment control significantly enhances the control flexibility and brings unlimited possibilities for the improvement of the delay robustness. Finally, we inspect the stability preservation problem when using the integer-order approximations for practical implementation.

Mechanisms of human dynamic object recognition revealed by sequential deep neural networks

Humans can quickly recognize objects in a dynamically changing world. This ability is showcased by the fact that observers succeed at recognizing objects in rapidly changing image sequences, at up to 13 ms/image. To date, the mechanisms that govern dynamic object recognition remain poorly understood. Here, we developed deep learning models for dynamic recognition and compared different computational mechanisms, contrasting feedforward and recurrent, single-image and sequential processing as well as different forms of adaptation. We found that only models that integrate images sequentially via lateral recurrence mirrored human performance (N = 36) and were predictive of trial-by-trial responses across image durations (13-80 ms/image). Importantly, models with sequential lateral-recurrent integration also captured how human performance changes as a function of image presentation durations, with models processing images for a few time steps capturing human object recognition at shorter presentation durations and models processing images for more time steps capturing human object recognition at longer presentation durations. Furthermore, augmenting such a recurrent model with adaptation markedly improved dynamic recognition performance and accelerated its representational dynamics, thereby predicting human trial-by-trial responses using fewer processing resources. Together, these findings provide new insights into the mechanisms rendering object recognition so fast and effective in a dynamic visual world.

Effect of fractional derivatives on amplitude chimeras and symmetry-breaking death states in networks of limit-cycle oscillators

We study networks of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators. We show that the networks exhibit diverse amplitude chimeras and oscillation death patterns. The occurrence of amplitude chimeras in a network of van der Pol oscillators is observed for the first time. A form of amplitude chimera, namely, "damped amplitude chimera" is observed and characterized, where the size of the incoherent region(s) increases continuously in the course of time, and the oscillations of drifting units are damped continuously until they are quenched to steady state. It is found that as the order of the fractional derivative decreases, the lifetime of classical amplitude chimeras increases, and there is a critical point at which there is a transition to damped amplitude chimeras. Overall, a decrease in the order of fractional derivatives reduces the propensity to synchronization and promotes oscillation death phenomena including solitary oscillation death and chimera death patterns that were unobserved in networks of integer-order oscillators. This effect of the fractional derivatives is verified by the stability analysis based on the properties of the master stability function of some collective dynamical states calculated from the block-diagonalized variational equations of the coupled systems. The present study generalizes the results of our recently studied network of fractional-order Stuart-Landau oscillators.

Multifaceted luminance gain control beyond photoreceptors in Drosophila

Animals navigating in natural environments must handle vast changes in their sensory input. Visual systems, for example, handle changes in luminance at many timescales, from slow changes across the day to rapid changes during active behavior. To maintain luminance-invariant perception, visual systems must adapt their sensitivity to changing luminance at different timescales. We demonstrate that luminance gain control in photoreceptors alone is insufficient to explain luminance invariance at both fast and slow timescales and reveal the algorithms that adjust gain past photoreceptors in the fly eye. We combined imaging and behavioral experiments with computational modeling to show that downstream of photoreceptors, circuitry taking input from the single luminance-sensitive neuron type L3 implements gain control at fast and slow timescales. This computation is bidirectional in that it prevents the underestimation of contrasts in low luminance and overestimation in high luminance. An algorithmic model disentangles these multifaceted contributions and shows that the bidirectional gain control occurs at both timescales. The model implements a nonlinear interaction of luminance and contrast to achieve gain correction at fast timescales and a dark-sensitive channel to improve the detection of dim stimuli at slow timescales. Together, our work demonstrates how a single neuronal channel performs diverse computations to implement gain control at multiple timescales that are together important for navigation in natural environments.

Diverse electrical responses in a network of fractional-order conductance-based excitable Morris-Lecar systems

The diverse excitabilities of cells often produce various spiking-bursting oscillations that are found in the neural system. We establish the ability of a fractional-order excitable neuron model with Caputo's fractional derivative to analyze the effects of its dynamics on the spike train features observed in our results. The significance of this generalization relies on a theoretical framework of the model in which memory and hereditary properties are considered. Employing the fractional exponent, we first provide information about the variations in electrical activities. We deal with the 2D class I and class II excitable Morris-Lecar (M-L) neuron models that show the alternation of spiking and bursting features including MMOs & MMBOs of an uncoupled fractional-order neuron. We then extend the study with the 3D slow-fast M-L model in the fractional domain. The considered approach establishes a way to describe various characteristics similarities between fractional-order and classical integer-order dynamics. Using the stability and bifurcation analysis, we discuss different parameter spaces where the quiescent state emerges in uncoupled neurons. We show the characteristics consistent with the analytical results. Next, the Erdös-Rényi network of desynchronized mixed neurons (oscillatory and excitable) is constructed that is coupled through membrane voltage. It can generate complex firing activities where quiescent neurons start to fire. Furthermore, we have shown that increasing coupling can create cluster synchronization, and eventually it can enable the network to fire in unison. Based on cluster synchronization, we develop a reduced-order model which can capture the activities of the entire network. Our results reveal that the effect of fractional-order depends on the synaptic connectivity and the memory trace of the system. Additionally, the dynamics captures spike frequency adaptation and spike latency that occur over multiple timescales as the effects of fractional derivative, which has been observed in neural computation.

Fractional order-induced bifurcations in a delayed neural network with three neurons

This paper reports the novel results on fractional order-induced bifurcation of a tri-neuron fractional-order neural network (FONN) with delays and instantaneous self-connections by the intersection of implicit function curves to solve the bifurcation critical point. Firstly, it considers the distribution of the root of the characteristic equation in depth. Subsequently, it views fractional order as the bifurcation parameter and establishes the transversal condition and stability interval. The main novelties of this paper are to systematically analyze the order as a bifurcation parameter and concretely establish the order critical value through an implicit function array, which is a novel idea to solve the critical value. The derived results exhibit that once the value of the fractional order is greater than the bifurcation critical value, the stability of the system will be smashed and Hopf bifurcation will emerge. Ultimately, the validity of the developed key fruits is elucidated via two numerical experiments.

Diverse nature of interictal oscillations: EEG-based biomarkers in epilepsy

Interictal electroencephalogram (EEG) patterns, including high-frequency oscillations (HFOs), interictal spikes (ISs), and slow wave activities (SWAs), are defined as specific oscillations between seizure events. These interictal oscillations reflect specific dynamic changes in network excitability and play various roles in epilepsy. In this review, we briefly describe the electrographic characteristics of HFOs, ISs, and SWAs in the interictal state, and discuss the underlying cellular and network mechanisms. We also summarize representative evidence from experimental and clinical epilepsy to address their critical roles in ictogenesis and epileptogenesis, indicating their potential as electrophysiological biomarkers of epilepsy. Importantly, we put forwards some perspectives for further research in the field.

Neural adaptation and fractional dynamics as a window to underlying neural excitability

The relationship between macroscale electrophysiological recordings and the dynamics of underlying neural activity remains unclear. We have previously shown that low frequency EEG activity (<1 Hz) is decreased at the seizure onset zone (SOZ), while higher frequency activity (1-50 Hz) is increased. These changes result in power spectral densities (PSDs) with flattened slopes near the SOZ, which are assumed to be areas of increased excitability. We wanted to understand possible mechanisms underlying PSD changes in brain regions of increased excitability. We hypothesized that these observations are consistent with changes in adaptation within the neural circuit. We developed a theoretical framework and tested the effect of adaptation mechanisms, such as spike frequency adaptation and synaptic depression, on excitability and PSDs using filter-based neural mass models and conductance-based models. We compared the contribution of single timescale adaptation and multiple timescale adaptation. We found that adaptation with multiple timescales alters the PSDs. Multiple timescales of adaptation can approximate fractional dynamics, a form of calculus related to power laws, history dependence, and non-integer order derivatives. Coupled with input changes, these dynamics changed circuit responses in unexpected ways. Increased input without synaptic depression increases broadband power. However, increased input with synaptic depression may decrease power. The effects of adaptation were most pronounced for low frequency activity (< 1Hz). Increased input combined with a loss of adaptation yielded reduced low frequency activity and increased higher frequency activity, consistent with clinical EEG observations from SOZs. Spike frequency adaptation and synaptic depression, two forms of multiple timescale adaptation, affect low frequency EEG and the slope of PSDs. These neural mechanisms may underlie changes in EEG activity near the SOZ and relate to neural hyperexcitability. Neural adaptation may be evident in macroscale electrophysiological recordings and provide a window to understanding neural circuit excitability.

Temporal derivative computation in the dorsal raphe network revealed by an experimentally driven augmented integrate-and-fire modeling framework

By means of an expansive innervation, the serotonin (5-HT) neurons of the dorsal raphe nucleus (DRN) are positioned to enact coordinated modulation of circuits distributed across the entire brain in order to adaptively regulate behavior. Yet the network computations that emerge from the excitability and connectivity features of the DRN are still poorly understood. To gain insight into these computations, we began by carrying out a detailed electrophysiological characterization of genetically identified mouse 5-HT and somatostatin (SOM) neurons. We next developed a single-neuron modeling framework that combines the realism of Hodgkin-Huxley models with the simplicity and predictive power of generalized integrate-and-fire models. We found that feedforward inhibition of 5-HT neurons by heterogeneous SOM neurons implemented divisive inhibition, while endocannabinoid-mediated modulation of excitatory drive to the DRN increased the gain of 5-HT output. Our most striking finding was that the output of the DRN encodes a mixture of the intensity and temporal derivative of its input, and that the temporal derivative component dominates this mixture precisely when the input is increasing rapidly. This network computation primarily emerged from prominent adaptation mechanisms found in 5-HT neurons, including a previously undescribed dynamic threshold. By applying a bottom-up neural network modeling approach, our results suggest that the DRN is particularly apt to encode input changes over short timescales, reflecting one of the salient emerging computations that dominate its output to regulate behavior.

A PID controller for synchronization between master-slave neurons in fractional-order of neocortical network model

Modeling of the biological neurons is a way to understand the architecture of neural networks of the brain. A complex brain network includes the synchronization between some groups of neurons. The dynamic behavior of interactions between groups of slave-master neurons in the neocortical network is unpredictable and challenging. The purpose of synchronizing a neural interaction is to reduce the synchronization error between the chaotic slave-master neurons. This paper uses a proportional-integral-derivative (PID) controller to synchronize master-slave neurons in the fractional-order of the neocortical network model based on dendritic spike frequency adaptation (DSFA) uncertainties and unknown disturbance effects. The purpose of this article is in two parts: First, we implemented the effect of previous states of the neuron conditions by fractional-order of the differential equations in the neocortical network model. Second, by synchronizing the FO neocortical master-slave model by PID controller, we investigated the connection strength of the complex network in chaotic point of view. The optimized PID coefficients and fractional-order were calculated using root mean square error (RMSE) criteria to control the membrane voltage synchronization. The chaotic behavior of the system was evaluated by numerical techniques such as attractor analysis and time series diagrams. The optimal RMSE value for master-slave neurons occurred at fractional-orders 0.89. It is shown that the synchronization of master-slave neurons improves over time, and eventually they are fully synchronized while the controller error is reduced.

Characterisation of visual guidance of steering to intercept targets following curving trajectories using Qualitative Inconsistency Detection

This study explored the informational variables guiding steering behaviour in a locomotor interception task with targets moving along circular trajectories. Using a new method of analysis focussing on the temporal co-evolution of steering behaviour and the potential information sources driving it, we set out to invalidate reliance on plausible informational candidates. Applied to individual trials rather than ensemble averages, this Qualitative Inconsistency Detection (QuID) method revealed that steering behaviour was not compatible with reliance on information grounded in any type of change in the agent-centred target-heading angle. First-order changes in the environment-centred target's bearing angle could also not adequately account for the variations in behaviour observed under the different experimental conditions. Capturing the observed timing of unfolding steering behaviour ultimately required a combination of (velocity-based) first-order and (acceleration-based) second-order changes in bearing angle. While this result may point to reliance on fractional-order based changes in bearing angle, the overall importance of the present findings resides in the demonstration of the necessity to break away from the existing practice of trying to fit behaviour into a priori postulated functional strategies based on categorical differences between operative heuristic rules or control laws.

Memoryless Optimality: Neurons Do Not Need Adaptation to Optimally Encode Stimuli With Arbitrarily Complex Statistics

Our neurons seem capable of handling any type of data, regardless of its scale or statistical properties. In this letter, we suggest that optimal coding may occur at the single-neuron level without requiring memory, adaptation, or evolutionary-driven fit to the stimuli. We refer to a neural circuit as optimal if it maximizes the mutual information between its inputs and outputs. We show that often encountered differentiator neurons, or neurons that respond mainly to changes in the input, are capable of using all their information capacity when handling samples of any statistical distribution. We demonstrate this optimality using both analytical methods and simulations. In addition to demonstrating the simplicity and elegance of neural processing, this result might provide a way to improve the handling of data by artificial neural networks.

Nonequilibrium dynamics of adaptation in sensory systems

Adaptation is used by biological sensory systems to respond to a wide range of environmental signals, by adapting their response properties to the statistics of the stimulus in order to maximize information transmission. We derive rules of optimal adaptation to changes in the mean and variance of a continuous stimulus in terms of Bayesian filters and map them onto stochastic equations that couple the state of the environment to an internal variable controlling the response function. We calculate numerical and exact results for the speed and accuracy of adaptation and its impact on information transmission. We find that, in the regime of efficient adaptation, the speed of adaptation scales sublinearly with the rate of change of the environment. Finally, we exploit the mathematical equivalence between adaptation and stochastic thermodynamics to quantitatively relate adaptation to the irreversibility of the adaptation time course, defined by the rate of entropy production. Our results suggest a means to empirically quantify adaptation in a model-free and nonparametric way.

Effects of double delays on bifurcation for a fractional-order neural network

Neural network bifurcation is an important nonlinear dynamic behavior of neural network, which plays an important role in cognitive calculation. The effects of leakage delay or communication delay on the stability and bifurcation of a fractional-order neural network (FONN) are researched. By viewing leakage delay or communication delay as the bifurcation parameters to detect the bifurcations conditions of the developed FONN, respectively, we capture the bifurcation points with regard to leakage delay or communication delay. It alleges that FONN exhibits excellent stability performance with choosing smaller values of them, and Hopf bifurcations emerge of FONN and induce poor performance if selecting a larger ones. In the end, numerical examples are employed to evaluate the feasibleness of the analytical discoveries.

Uniform Stability of Complex-Valued Neural Networks of Fractional Order With Linear Impulses and Fixed Time Delays

As a generation of the real-valued neural network (RVNN), complex-valued neural network (CVNN) is based on the complex-valued (CV) parameters and variables. The fractional-order (FO) CVNN with linear impulses and fixed time delays is discussed. By using the sign function, the Banach fixed point theorem, and two classes of activation functions, some criteria of uniform stability for the solution and existence and uniqueness for equilibrium solution are derived. Finally, three experimental simulations are presented to illustrate the correctness and effectiveness of the obtained results.

Implementation of synchronization of multi-fractional-order of chaotic neural networks with a variety of multi-time-delays: Studying the effect of double encryption for text encryption

This research proposes the idea of double encryption, which is the combination of chaos synchronization of non-identical multi-fractional-order neural networks with multi-time-delays (FONNSMD) and symmetric encryption. Symmetric encryption is well known to be outstanding in speed and accuracy but less effective. Therefore, to increase the strength of data protection effectively, we combine both methods where the secret keys are generated from the third part of the neural network systems (NNS) and used only once to encrypt and decrypt the message. In addition, a fractional-order Lyapunov direct function (FOLDF) is designed and implemented in sliding mode control systems (SMCS) to maintain the convergence of approximated synchronization errors. Finally, three examples are carried out to confirm the theoretical analysis and find which synchronization is achieved. Then the result is combined with symmetric encryption to increase the security of secure communication, and a numerical simulation verifies the method's accuracy.

Stability of Fractional-Order Quasi-Linear Impulsive Integro-Differential Systems with Multiple Delays

In this paper, some novel conditions for the stability results for a class of fractional-order quasi-linear impulsive integro-differential systems with multiple delays is discussed. First, the existence and uniqueness of mild solutions for the considered system is discussed using contraction mapping theorem. Then, novel conditions for Mittag–Leffler stability (MLS) of the considered system are established by using well known mathematical techniques, and further, the two corollaries are deduced, which still gives some new results. Finally, an example is given to illustrate the applications of the results.

The emergence of a collective sensory response threshold in ant colonies

SignificanceIn this study, we ask how ant colonies integrate information about the external environment with internal state parameters to produce adaptive, system-level responses. First, we show that colonies collectively evacuate the nest when the ground temperature becomes too warm. The threshold temperature for this response is a function of colony size, with larger colonies evacuating the nest at higher temperatures. The underlying dynamics can thus be interpreted as a decision-making process that takes both temperature (external environment) and colony size (internal state) into account. Using mathematical modeling, we show that these dynamics can emerge from a balance between local excitatory and global inhibitory forces acting between the ants. Our findings in ants parallel other complex biological systems like neural circuits.

Uniform Stability of a Class of Fractional-Order Fuzzy Complex-Valued Neural Networks in Infinite Dimensions

In this paper, the problem of the uniform stability for a class of fractional-order fuzzy impulsive complex-valued neural networks with mixed delays in infinite dimensions is discussed for the first time. By utilizing fixed-point theory, theory of differential inclusion and set-valued mappings, the uniqueness of the solution of the above complex-valued neural networks is derived. Subsequently, the criteria for uniform stability of the above complex-valued neural networks are established. In comparison with related results, we do not need to construct a complex Lyapunov function, reducing the computational complexity. Finally, an example is given to show the validity of the main results.

Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model

The aim of this work is to describe the dynamics of a fractional-order coupled FitzHugh–Nagumo neuronal model. The equilibrium states are analyzed in terms of their stability properties, both dependently and independently of the fractional orders of the Caputo derivatives, based on recently established theoretical results. Numerical simulations are shown to clarify and exemplify the theoretical results.

Synchronization of Fractional Order Neurons in Presence of Noise

The firing rate of some biological neurons such as neocortical pyramidal neurons is consistent with fractional order derivative, and the fractional-order neuron models depict the firing rate of neurons more accurately than other integer order neuron models do. For this reason, first, the dynamical characteristics of fractional order Hindmarsh Rose (HR) neuron are investigated, here and then a two coupled neuronal system based on Hindmarsh Rose neuron is presented. The results show several differences in the dynamical cha.racteristics of integer order and fractional order Hindmarsh Rose neuron model. The integer order model shows only one type of firing characteristics when the parameter of the model remained the same. The fractional-order model depicts several dynamical behaviors even for the same parameters as the order of the fractional operator is varied with the same parameter values. The firing frequency increases as the order of the fractional operator decreases. The fractional-order is therefore key in determining the firing characteristics of biological neuron models. A linearized model of HR neuron is also given for hardware resource minimizations and to implement this neuronal network on a large scale. A synchronized system of two fractional-order fractional Hindmarsh-Rose (HR) neurons in the presence of noise is also presented. The dynamical characteristics of the modified coupled neuron are determined by the parameters of the neuron model and the coupling function. The robustness of the network in the presence of noise is verified by both amplitude and phase synchronization techniques. A simplification of the coupling function is also presented to reduce the hardware cost. The synchronization results show that the model can produce the desired behavior with acceptable error.

Global Exponential Stability of Fractional Order Complex-Valued Neural Networks with Leakage Delay and Mixed Time Varying Delays

This paper investigates the global exponential stability of fractional order complex-valued neural networks with leakage delay and mixed time varying delays. By constructing a proper Lyapunov-functional we established sufficient conditions to ensure global exponential stability of the fractional order complex-valued neural networks. The stability conditions are established in terms of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

A new biological central pattern generator model and its relationship with the motor units

The central pattern generator (CPG) is a key neural-circuit component of the locomotion control system. Recently, numerous molecular and genetic approaches have been proposed for investigating the CPG mechanisms. The rhythm in the CPG locomotor circuits comes from the activity in the ipsilateral excitatory neurons whose output is controlled by inter-neuron inhibitory connections. Conventional models for simulating the CPG mechanism are complex Hodgkin-Huxley-type models. Inspired by biological investigations and the continuous-time Matsuoka model, we propose new integral-order and fractional-order CPG models, which consider time delays and synaptic interfaces. The phase diagrams exhibit limit cycles and periodic solutions, in agreement with the CPG biological characteristics. As well, the fractional-order model shows state transitions with order variations. In addition, we investigate the relationship between the CPG and the motor units through the construction of integral-order and fractional-order coupling models. Simulations of these coupling models show that the states generated by the three motor units are in accordance with the experimentally-obtained states in previous studies. The proposed models reveal that the CPG can regulate limb locomotion patterns through connection weights and synaptic interfaces. Moreover, in comparison to the integral-order models, the fractional-order ones appear to be more effective, and hence more suitable for describing the dynamics of the CPG biological system.

Improved Results on Finite-Time Passivity and Synchronization Problem for Fractional-Order Memristor-Based Competitive Neural Networks: Interval Matrix Approach

This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust control scheme cannot guarantee the synchronization of FOMBCNNs. Under the framework of the Filippov solution, the drive and response FOMBCNNs are first transformed into systems with interval parameters. Then, the new sufficient criteria are obtained by linear matrix inequalities (LMIs) to ensure the passivity in finite-time criteria for FOMBCNNs with mismatched switching jumps. Further, a feedback control law is designed to ensure the finite-time synchronization of FOMBCNNs. Finally, three numerical cases are given to illustrate the usefulness of our passivity and synchronization results.