Fractal ion-channel behavior generates fractal firing patterns in neuronal models

Dynamics of neuronal firing modulated by high-frequency electrical pulse stimulations at axons in rat hippocampus

Objective. The development of electrical pulse stimulations in brain, including deep brain stimulation, is promising for treating various brain diseases. However, the mechanisms of brain stimulations are not yet fully understood. Previous studies have shown that the commonly used high-frequency stimulation (HFS) can increase the firing of neurons and modulate the pattern of neuronal firing. Because the generation of neuronal firing in brain is a nonlinear process, investigating the characteristics of nonlinear dynamics induced by HFS could be helpful to reveal more mechanisms of brain stimulations. The aim of present study is to investigate the fractal properties in the neuronal firing generated by HFS.Approach. HFS pulse sequences with a constant frequency 100 Hz were applied in the afferent fiber tracts of rat hippocampal CA1 region. Unit spikes of both the pyramidal cells and the interneurons in the downstream area of stimulations were recorded. Two fractal indexes-the Fano factor and Hurst exponent were calculated to evaluate the changes of long-range temporal correlations (LRTCs), a typical characteristic of fractal process, in spike sequences of neuronal firing.Mainresults. Neuronal firing at both baseline and during HFS exhibited LRTCs over multiple time scales. In addition, the LRTCs significantly increased during HFS, which was confirmed by simulation data of both randomly shuffled sequences and surrogate sequences.Conclusion. The purely periodic stimulation of HFS pulses, a non-fractal process without LRTCs, can increase rather than decrease the LRTCs in neuronal firing.Significance. The finding provides new nonlinear mechanisms of brain stimulation and suggests that LRTCs could be a new biomarker to evaluate the nonlinear effects of HFS.

The fractal brain: scale-invariance in structure and dynamics

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

Emergence of lump-like solitonic waves in Heimburg-Jackson biomembranes and nerves fractal model

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev-Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

Excessive and less complex body movement in children with autism during face-to-face conversation: An objective approach to behavioral quantification

The majority of existing studies investigating characteristics of overt social behavior in individuals with autism spectrum disorder (ASD) relied on informants' evaluation through questionnaires and behavioral coding techniques. As a novelty, this study aimed to quantify the complex movements produced during social interactions in order to test differences in ASD movement dynamics and their convergence, or lack thereof, during social interactions. Twenty children with ASD and twenty-three children with typical development (TD) were videotaped while engaged in a face-to-face conversation with an interviewer. An image differencing technique was utilized to extract the movement time series. Spectral analyses were conducted to quantify the average power of movement, and the fractal scaling of movement. The degree of complexity matching was calculated to capture the level of behavioral coordination between the interviewer and children. Results demonstrated that the average power was significantly higher (p < 0.01), and the fractal scaling was steeper (p < 0.05) in children with ASD, suggesting excessive and less complex movement as compared to the TD peers. Complexity matching occurred between children and interviewers, but there was no reliable difference in the strength of matching between the ASD and TD children. Descriptive trends in the interviewer's behavior suggest that her movements adapted to match both ASD and TD movements equally well. The findings of our study might shed light on seeking novel behavioral markers of ASD, and on developing automatic ASD screening techniques during daily social interactions.

Neural Network Differential Equations For Ion Channel Modelling

Mathematical models of cardiac ion channels have been widely used to study and predict the behaviour of ion currents. Typically models are built using biophysically-based mechanistic principles such as Hodgkin-Huxley or Markov state transitions. These models provide an abstract description of the underlying conformational changes of the ion channels. However, due to the abstracted conformation states and assumptions for the rates of transition between them, there are differences between the models and reality-termed model discrepancy or misspecification. In this paper, we demonstrate the feasibility of using a mechanistically-inspired neural network differential equation model, a hybrid non-parametric model, to model ion channel kinetics. We apply it to the hERG potassium ion channel as an example, with the aim of providing an alternative modelling approach that could alleviate certain limitations of the traditional approach. We compare and discuss multiple ways of using a neural network to approximate extra hidden states or alternative transition rates. In particular we assess their ability to learn the missing dynamics, and ask whether we can use these models to handle model discrepancy. Finally, we discuss the practicality and limitations of using neural networks and their potential applications.

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.

Long-Term Activity Dynamics of Single Neurons and Networks

The firing rate of neuronal spiking in vitro and in vivo significantly varies over extended timescales, characterized by long-memory processes and complex statistics, and appears in spontaneous as well as evoked activity upon repeated stimulus presentation. These variations in response features and their statistics, in face of repeated instances of a given physical input, are ubiquitous in all levels of brain-behavior organization. They are expressed in single neuron and network response variability but even appear in variations of subjective percepts or psychophysical choices and have been described as stemming from history-dependent, stochastic, or rate-determined processes.But what are the sources underlying these temporally rich variations in firing rate? Are they determined by interactions of the nervous system as a whole, or do isolated, single neurons or neuronal networks already express these fluctuations independent of higher levels? These questions motivated the application of a method that allows for controlled and specific long-term activation of a single neuron or neuronal network, isolated from higher levels of cortical organization.This chapter highlights the research done in cultured cortical networks to study (1) the inherent non-stationarity of neuronal network activity, (2) single neuron response fluctuations and underlying processes, and (3) the interface layer between network and single cell, the non-stationary efficacy of the ensemble of synapses impinging onto the observed neuron.

Traits of criticality in membrane potential fluctuations of pyramidal neurons in the CA1 region of rat hippocampus

Evidence that neural circuits are operating near criticality has been provided at various levels of brain organisation with a presumed role in maximising information processing and multiscale activity association. Criticality has been linked to excitation at both the single-cell and network levels, as action potential generation marks an obvious phase transition from a resting to an excitable state. Using in vitro intracellular recordings, we examine irregular, small amplitude membrane potential fluctuations from CA1 pyramidal neurons of Wistar male rats. We show that these fluctuations, confounded with noise, carry information relevant to the neuronal state. The underlying dynamics exhibit intermittent characteristics shown to describe the temporal fluctuations of the order parameter of a macroscopic system at its critical point even in the absence of firing. An externally applied stimulus serves as the control parameter, driving the system in and out of its critical state. Based on our experimental observations we calculate the equivalent of the isothermal critical exponent δ_h finding a value which depends on the applied stimulus. For each neuron there is a stimulus amplitude for which the critical behaviour becomes most pronounced. The corresponding mean value of δ_h in the considered ensemble of neurons is δ_h  ≈ 1.89, close to theoretical predictions for critical networks. Finally, we show that the firing rate of a neuron decreases exponentially with δ_h .

Hearing the Sound in the Brain: Influences of Different EEG References

If the scalp potential signals, the electroencephalogram (EEG), are due to neural "singers" in the brain, how could we listen to them with less distortion? One crucial point is that the data recording on the scalp should be faithful and accurate, thus the choice of reference electrode is a vital factor determining the faithfulness of the data. In this study, music on the scalp derived from data in the brain using three different reference electrodes were compared, including approximate zero reference-reference electrode standardization technique (REST), average reference (AR), and linked mastoids reference (LM). The classic music pieces in waveform format were used as simulated sources inside a head model, and they were forward calculated to scalp as standard potential recordings, i.e., waveform format music from the brain with true zero reference. Then these scalp music was re-referenced into REST, AR, and LM based data, and compared with the original forward data (true zero reference). For real data, the EEG recorded in an orthodontic pain control experiment were utilized for music generation with the three references, and the scale free index (SFI) of these music pieces were compared. The results showed that in the simulation for only one source, different references do not change the music/waveform; for two sources or more, REST provide the most faithful music/waveform to the original ones inside the brain, and the distortions caused by AR and LM were spatial locations of both source and scalp electrode dependent. The brainwave music from the real EEG data showed that REST and AR make the differences of SFI between two states more recognized and found the frontal is the main region that producing the music. In conclusion, REST can reconstruct the true signals approximately, and it can be used to help to listen to the true voice of the neural singers in the brain.

Dual-induced multifractality in online viewing activity

Although recent studies have found that the long-term correlations relating to the fat-tailed distribution of inter-event times exist in human activity and that these correlations indicate the presence of fractality, the property of fractality and its origin have not been analyzed. We use both detrended fluctuation analysis and multifractal detrended fluctuation analysis to analyze the time series in online viewing activity separating from Movielens and Netflix. We find long-term correlations at both the individual and communal levels and that the extent of correlation at the individual level is determined by the activity level. These long-term correlations also indicate that there is fractality in the pattern of online viewing. We first find a multifractality that results from the combined effect of the fat-tailed distribution of inter-event times (i.e., the times between successive viewing actions of individuals) and the long-term correlations in online viewing activity and verify this finding using three synthesized series. Therefore, it can be concluded that the multifractality in online viewing activity is caused by both the fat-tailed distribution of inter-event times and the long-term correlations and that this enlarges the generic property of human activity to include not just physical space but also cyberspace.

Regularity of beating of small clusters of embryonic chick ventricular heart-cells: experiment vs. stochastic single-channel population model

The transmembrane potential is recorded from small isopotential clusters of 2-4 embryonic chick ventricular cells spontaneously generating action potentials. We analyze the cycle-to-cycle fluctuations in the time between successive action potentials (the interbeat interval or IBI). We also convert an existing model of electrical activity in the cluster, which is formulated as a Hodgkin-Huxley-like deterministic system of nonlinear ordinary differential equations describing five individual ionic currents, into a stochastic model consisting of a population of ∼20 000 independently and randomly gating ionic channels, with the randomness being set by a real physical stochastic process (radio static). This stochastic model, implemented using the Clay-DeFelice algorithm, reproduces the fluctuations seen experimentally: e.g., the coefficient of variation (standard deviation/mean) of IBI is 4.3% in the model vs. the 3.9% average value of the 17 clusters studied. The model also replicates all but one of several other quantitative measures of the experimental results, including the power spectrum and correlation integral of the voltage, as well as the histogram, Poincaré plot, serial correlation coefficients, power spectrum, detrended fluctuation analysis, approximate entropy, and sample entropy of IBI. The channel noise from one particular ionic current (I_Ks), which has channel kinetics that are relatively slow compared to that of the other currents, makes the major contribution to the fluctuations in IBI. Reproduction of the experimental coefficient of variation of IBI by adding a Gaussian white noise-current into the deterministic model necessitates using an unrealistically high noise-current amplitude. Indeed, a major implication of the modelling results is that, given the wide range of time-scales over which the various species of channels open and close, only a cell-specific stochastic model that is formulated taking into consideration the widely different ranges in the frequency content of the channel-noise produced by the opening and closing of several different types of channels will be able to reproduce precisely the various effects due to membrane noise seen in a particular electrophysiological preparation.

Memory in a fractional-order cardiomyocyte model alters properties of alternans and spontaneous activity

Cardiac memory is the dependence of electrical activity on the prior history of one or more system state variables, including transmembrane potential (V_m), ionic current gating, and ion concentrations. While prior work has represented memory either phenomenologically or with biophysical detail, in this study, we consider an intermediate approach of a minimal three-variable cardiomyocyte model, modified with fractional-order dynamics, i.e., a differential equation of order between 0 and 1, to account for history-dependence. Memory is represented via both capacitive memory, due to fractional-order V_m dynamics, that arises due to non-ideal behavior of membrane capacitance; and ionic current gating memory, due to fractional-order gating variable dynamics, that arises due to gating history-dependence. We perform simulations for varying V_m and gating variable fractional-orders and pacing cycle length and measure action potential duration (APD) and incidence of alternans, loss of capture, and spontaneous activity. In the absence of ionic current gating memory, we find that capacitive memory, i.e., decreased V_m fractional-order, typically shortens APD, suppresses alternans, and decreases the minimum cycle length (MCL) for loss of capture. However, in the presence of ionic current gating memory, capacitive memory can prolong APD, promote alternans, and increase MCL. Further, we find that reduced V_m fractional order (typically less than 0.75) can drive phase 4 depolarizations that promote spontaneous activity. Collectively, our results demonstrate that memory reproduced by a fractional-order model can play a role in alternans formation and pacemaking, and in general, can greatly increase the range of electrophysiological characteristics exhibited by a minimal model.

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.

Fractal analysis reveals subclasses of neurons and suggests an explanation of their spontaneous activity

The present work used fractal time series analysis (detrended fluctuation analysis; DFA) to examine the spontaneous activity of single neurons in an anesthetized animal model, specifically, the mitral cells in the rat main olfactory bulb. DFA bolstered previous research in suggesting two subclasses of mitral cells. Although there was no difference in the fractal scaling of the interspike interval series at the shorter timescales, there was a significant difference at longer timescales. Neurons in Group B exhibited fractal, power-law scaled interspike intervals, whereas neurons in Group A exhibited random variation. These results raise questions about the role of these different cells within the olfactory bulb and potential explanations of their dynamics. Specifically, self-organized criticality has been proposed as an explanation of fractal scaling in many natural systems, including neural systems. However, this theory is based on certain assumptions that do not clearly hold in the case of spontaneous neural activity, which likely reflects intrinsic cell dynamics rather than activity driven by external stimulation. Moreover, it is unclear how self-organized criticality might account for the random dynamics observed in Group A, and how these random dynamics might serve some functional role when embedded in the typical activity of the olfactory bulb. These theoretical considerations provide direction for additional experimental work.

ModFossa: A library for modeling ion channels using Python

The creation and simulation of ion channel models using continuous-time Markov processes is a powerful and well-used tool in the field of electrophysiology and ion channel research. While several software packages exist for the purpose of ion channel modeling, most are GUI based, and none are available as a Python library. In an attempt to provide an easy-to-use, yet powerful Markov model-based ion channel simulator, we have developed ModFossa, a Python library supporting easy model creation and stimulus definition, complete with a fast numerical solver, and attractive vector graphics plotting.

Complexity matching effects in bimanual and interpersonal syncopated finger tapping

The current study was designed to investigate complexity matching during syncopated behavioral coordination. Participants either tapped in (bimanual) syncopation using their two hands, or tapped in (interpersonal) syncopation with a partner, with each participant using one of their hands. The time series of inter-tap intervals (ITI) from each hand were submitted to fractal analysis, as well as to short-term and multi-timescale cross-correlation analyses. The results demonstrated that the fractal scaling of one hand's ITI was strongly correlated to that of the other hand, and this complexity matching effect was stronger in the bimanual condition than in the interpersonal condition. Moreover, the degree of complexity matching was predicted by the strength of short-term cross-correlation and the stability of the asynchrony between the two tapping series. These results suggest that complexity matching is not specific to the inphase synchronization tasks used in past research, but is a general result of coordination between complex systems.

Accurate Langevin approaches to simulate Markovian channel dynamics

The stochasticity of ion-channels dynamic is significant for physiological processes on neuronal cell membranes. Microscopic simulations of the ion-channel gating with Markov chains can be considered to be an accurate standard. However, such Markovian simulations are computationally demanding for membrane areas of physiologically relevant sizes, which makes the noise-approximating or Langevin equation methods advantageous in many cases. In this review, we discuss the Langevin-like approaches, including the channel-based and simplified subunit-based stochastic differential equations proposed by Fox and Lu, and the effective Langevin approaches in which colored noise is added to deterministic differential equations. In the framework of Fox and Lu's classical models, several variants of numerical algorithms, which have been recently developed to improve accuracy as well as efficiency, are also discussed. Through the comparison of different simulation algorithms of ion-channel noise with the standard Markovian simulation, we aim to reveal the extent to which the existing Langevin-like methods approximate results using Markovian methods. Open questions for future studies are also discussed.

Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model

Excitable cells and cell membranes are often modeled by the simple yet elegant parallel resistor-capacitor circuit. However, studies have shown that the passive properties of membranes may be more appropriately modeled with a non-ideal capacitor, in which the current-voltage relationship is given by a fractional-order derivative. Fractional-order membrane potential dynamics introduce capacitive memory effects, i.e., dynamics are influenced by a weighted sum of the membrane potential prior history. However, it is not clear to what extent fractional-order dynamics may alter the properties of active excitable cells. In this study, we investigate the spiking properties of the neuronal membrane patch, nerve axon, and neural networks described by the fractional-order Hodgkin-Huxley neuron model. We find that in the membrane patch model, as fractional-order decreases, i.e., a greater influence of membrane potential memory, peak sodium and potassium currents are altered, and spike frequency and amplitude are generally reduced. In the nerve axon, the velocity of spike propagation increases as fractional-order decreases, while in a neural network, electrical activity is more likely to cease for smaller fractional-order. Importantly, we demonstrate that the modulation of the peak ionic currents that occurs for reduced fractional-order alone fails to reproduce many of the key alterations in spiking properties, suggesting that membrane capacitive memory and fractional-order membrane potential dynamics are important and necessary to reproduce neuronal electrical activity.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Multifractal analysis of information processing in hippocampal neural ensembles during working memory under Δ⁹-tetrahydrocannabinol administration

BACKGROUND

Multifractal analysis quantifies the time-scale-invariant properties in data by describing the structure of variability over time. By applying this analysis to hippocampal interspike interval sequences recorded during performance of a working memory task, a measure of long-range temporal correlations and multifractal dynamics can reveal single neuron correlates of information processing.

NEW METHOD

Wavelet leaders-based multifractal analysis (WLMA) was applied to hippocampal interspike intervals recorded during a working memory task. WLMA can be used to identify neurons likely to exhibit information processing relevant to operation of brain-computer interfaces and nonlinear neuronal models.

RESULTS

Neurons involved in memory processing ("Functional Cell Types" or FCTs) showed a greater degree of multifractal firing properties than neurons without task-relevant firing characteristics. In addition, previously unidentified FCTs were revealed because multifractal analysis suggested further functional classification. The cannabinoid type-1 receptor (CB1R) partial agonist, tetrahydrocannabinol (THC), selectively reduced multifractal dynamics in FCT neurons compared to non-FCT neurons.

COMPARISON WITH EXISTING METHODS

WLMA is an objective tool for quantifying the memory-correlated complexity represented by FCTs that reveals additional information compared to classification of FCTs using traditional z-scores to identify neuronal correlates of behavioral events.

CONCLUSION

z-Score-based FCT classification provides limited information about the dynamical range of neuronal activity characterized by WLMA. Increased complexity, as measured with multifractal analysis, may be a marker of functional involvement in memory processing. The level of multifractal attributes can be used to differentially emphasize neural signals to improve computational models and algorithms underlying brain-computer interfaces.

Synaptic dynamics contribute to long-term single neuron response fluctuations

Firing rate variability at the single neuron level is characterized by long-memory processes and complex statistics over a wide range of time scales (from milliseconds up to several hours). Here, we focus on the contribution of non-stationary efficacy of the ensemble of synapses–activated in response to a given stimulus–on single neuron response variability. We present and validate a method tailored for controlled and specific long-term activation of a single cortical neuron in vitro via synaptic or antidromic stimulation, enabling a clear separation between two determinants of neuronal response variability: membrane excitability dynamics vs. synaptic dynamics. Applying this method we show that, within the range of physiological activation frequencies, the synaptic ensemble of a given neuron is a key contributor to the neuronal response variability, long-memory processes and complex statistics observed over extended time scales. Synaptic transmission dynamics impact on response variability in stimulation rates that are substantially lower compared to stimulation rates that drive excitability resources to fluctuate. Implications to network embedded neurons are discussed.

The neuronal response at extended timescales: long-term correlations without long-term memory

Long term temporal correlations frequently appear at many levels of neural activity. We show that when such correlations appear in isolated neurons, they indicate the existence of slow underlying processes and lead to explicit conditions on the dynamics of these processes. Moreover, although these slow processes can potentially store information for long times, we demonstrate that this does not imply that the neuron possesses a long memory of its input, even if these processes are bidirectionally coupled with neuronal response. We derive these results for a broad class of biophysical neuron models, and then fit a specific model to recent experiments. The model reproduces the experimental results, exhibiting long term (days-long) correlations due to the interaction between slow variables and internal fluctuations. However, its memory of the input decays on a timescale of minutes. We suggest experiments to test these predictions directly.

Self-organized criticality in single-neuron excitability

We present experimental and theoretical arguments, at the single-neuron level, suggesting that neuronal response fluctuations reflect a process that positions the neuron near a transition point that separates excitable and unexcitable phases. This view is supported by the dynamical properties of the system as observed in experiments on isolated cultured cortical neurons, as well as by a theoretical mapping between the constructs of self-organized criticality and membrane excitability biophysics.

Temporal whitening by power-law adaptation in neocortical neurons

Spike-frequency adaptation (SFA) is widespread in the CNS, but its function remains unclear. In neocortical pyramidal neurons, adaptation manifests itself by an increase in the firing threshold and by adaptation currents triggered after each spike. Combining electrophysiological recordings in mice with modeling, we found that these adaptation processes lasted for more than 20 s and decayed over multiple timescales according to a power law. The power-law decay associated with adaptation mirrored and canceled the temporal correlations of input current received in vivo at the somata of layer 2/3 somatosensory pyramidal neurons. These findings suggest that, in the cortex, SFA causes temporal decorrelation of output spikes (temporal whitening), an energy-efficient coding procedure that, at high signal-to-noise ratio, improves the information transfer.

Channel-based Langevin approach for the stochastic Hodgkin-Huxley neuron

Stochasticity in ion channel gating is the major source of intrinsic neuronal noise, which can induce many important effects in neuronal dynamics. Several numerical implementations of the Langevin approach have been proposed to approximate the Markovian dynamics of the Hodgkin-Huxley neuronal model. In this work an improved channel-based Langevin approach is proposed by introducing a truncation procedure to limit the state fractions in the range of [0, 1]. The truncated fractions are put back into the state fractions in the next time step for channel noise calculation. Our simulations show that the bounded Langevin approaches combined with the restored process give better approximations to the statistics of action potentials with the Markovian method. As a result, in our approach the channel state fractions are disturbed by two terms of noise: an uncorrelated Gaussian noise and a time-correlated noise obtained from the truncated fractions. We suggest that the restoration of truncated fractions is a critical process for a bounded Langevin method.

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic environments. The fold 'conotoxins' (peptides) could be unambiguously identified as one type with the least stability. The same analyses revealed that peptide-dipoles in the α/β class of proteins, in general, are more correlated to each other than are the peptide-dipoles in proteins belonging to the all-α class. Highly favorable electrostatic milieu in the interiors of TIM-barrel, α/β-hydrolase structures could explain their remarkably conserved (evolutionary) stability from a new light. Finally, we point out certain inherent limitations of fractal constructs before attempting to identify the areas and problems where the implementation of fractal dimension-based constructs can be of paramount help to unearth latent information on protein structural properties.

Dynamics of excitability over extended timescales in cultured cortical neurons

Although neuronal excitability is well understood and accurately modeled over timescales of up to hundreds of milliseconds, it is currently unclear whether extrapolating from this limited duration to longer behaviorally relevant timescales is appropriate. Here we used an extracellular recording and stimulation paradigm that extends the duration of single-neuron electrophysiological experiments, exposing the dynamics of excitability in individual cultured cortical neurons over timescales hitherto inaccessible. We show that the long-term neuronal excitability dynamics is unstable and dominated by critical fluctuations, intermittency, scale-invariant rate statistics, and long memory. These intrinsic dynamics bound the firing rate over extended timescales, contrasting observed short-term neuronal response to stimulation onset. Furthermore, the activity of a neuron over extended timescales shows transitions between quasi-stable modes, each characterized by a typical response pattern. Like in the case of rate statistics, the short-term onset response pattern that often serves to functionally define a given neuron is not indicative of its long-term ongoing response. These observations question the validity of describing neuronal excitability based on temporally restricted electrophysiological data, calling for in-depth exploration of activity over wider temporal scales. Such extended experiments will probably entail a different kind of neuronal models, accounting for the unbounded range, from milliseconds up.

Chipping away at memory

Inverse power-law behavior is known to be characteristic of adaptation, learning, and memory. Herein, we propose a phenomenological model of forgetting based on renewal theory that introduces a new psychophysical concept, chipping; discrete events that chip away at chunks of memory and thereby produce forgetting. The neural mechanism producing these chips is the 1/f-noise that is generically produced in complex neuronal networks.

Fractals in the nervous system: conceptual implications for theoretical neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

History-dependent Dynamics in a Generic Model of Ion Channels - an Analytic Study

Recent experiments have demonstrated that the timescale of adaptation of single neurons and ion channel populations to stimuli slows down as the length of stimulation increases; in fact, no upper bound on temporal timescales seems to exist in such systems. Furthermore, patch clamp experiments on single ion channels have hinted at the existence of large, mostly unobservable, inactivation state spaces within a single ion channel. This raises the question of the relation between this multitude of inactivation states and the observed behavior. In this work we propose a minimal model for ion channel dynamics which does not assume any specific structure of the inactivation state space. The model is simple enough to render an analytical study possible. This leads to a clear and concise explanation of the experimentally observed exponential history-dependent relaxation in sodium channels in a voltage clamp setting, and shows that their recovery rate from slow inactivation must be voltage dependent. Furthermore, we predict that history-dependent relaxation cannot be created by overly sparse spiking activity. While the model was created with ion channel populations in mind, its simplicity and genericalness render it a good starting point for modeling similar effects in other systems, and for scaling up to higher levels such as single neurons which are also known to exhibit multiple time scales.

Characterization of intermittency in renewal processes: application to earthquakes

We construct a one-dimensional piecewise linear intermittent map from the interevent time distribution for a given renewal process. Then, we characterize intermittency by the asymptotic behavior near the indifferent fixed point in the piecewise linear intermittent map. Thus, we provide a framework to understand a unified characterization of intermittency and also present the Lyapunov exponent for renewal processes. This method is applied to the occurrence of earthquakes using the Japan Meteorological Agency and the National Earthquake Information Center catalog. By analyzing the return map of interevent times, we find that interevent times are not independent and identically distributed random variables but that the conditional probability distribution functions in the tail obey the Weibull distribution.

Scale-free music of the brain

Background

There is growing interest in the relation between the brain and music. The appealing similarity between brainwaves and the rhythms of music has motivated many scientists to seek a connection between them. A variety of transferring rules has been utilized to convert the brainwaves into music; and most of them are mainly based on spectra feature of EEG.

Methodology/Principal Findings

In this study, audibly recognizable scale-free music was deduced from individual Electroencephalogram (EEG) waveforms. The translation rules include the direct mapping from the period of an EEG waveform to the duration of a note, the logarithmic mapping of the change of average power of EEG to music intensity according to the Fechner's law, and a scale-free based mapping from the amplitude of EEG to music pitch according to the power law. To show the actual effect, we applied the deduced sonification rules to EEG segments recorded during rapid-eye movement sleep (REM) and slow-wave sleep (SWS). The resulting music is vivid and different between the two mental states; the melody during REM sleep sounds fast and lively, whereas that in SWS sleep is slow and tranquil. 60 volunteers evaluated 25 music pieces, 10 from REM, 10 from SWS and 5 from white noise (WN), 74.3% experienced a happy emotion from REM and felt boring and drowsy when listening to SWS, and the average accuracy for all the music pieces identification is 86.8%(κ = 0.800, P<0.001). We also applied the method to the EEG data from eyes closed, eyes open and epileptic EEG, and the results showed these mental states can be identified by listeners.

Conclusions/Significance

The sonification rules may identify the mental states of the brain, which provide a real-time strategy for monitoring brain activities and are potentially useful to neurofeedback therapy.

Levels of complexity in scale-invariant neural signals

Many physical and physiological signals exhibit complex scale-invariant features characterized by 1/f scaling and long-range power-law correlations, indicating a possibly common control mechanism. Specifically, it has been suggested that dynamical processes, influenced by inputs and feedback on multiple time scales, may be sufficient to give rise to 1/f scaling and scale invariance. Two examples of physiologic signals that are the output of hierarchical multiscale physiologic systems under neural control are the human heartbeat and human gait. Here we show that while both cardiac interbeat interval and gait interstride interval time series under healthy conditions have comparable 1/f scaling, they still may belong to different complexity classes. Our analysis of the multifractal scaling exponents of the fluctuations in these two signals demonstrates that in contrast to the multifractal behavior found in healthy heartbeat dynamics, gait time series exhibit less complex, close to monofractal behavior. Further, we find strong anticorrelations in the sign and close to random behavior for the magnitude of gait fluctuations at short and intermediate time scales, in contrast to weak anticorrelations in the sign and strong positive correlation for the magnitude of heartbeat interval fluctuations-suggesting that the neural mechanisms of cardiac and gait control exhibit different linear and nonlinear features. These findings are of interest because they underscore the limitations of traditional two-point correlation methods in fully characterizing physiological and physical dynamics. In addition, these results suggest that different mechanisms of control may be responsible for varying levels of complexity observed in physiological systems under neural regulation and in physical systems that possess similar 1/f scaling.

Adaptive transition rates in excitable membranes

Adaptation of activity in excitable membranes occurs over a wide range of timescales. Standard computational approaches handle this wide temporal range in terms of multiple states and related reaction rates emanating from the complexity of ionic channels. The study described here takes a different (perhaps complementary) approach, by interpreting ion channel kinetics in terms of population dynamics. I show that adaptation in excitable membranes is reducible to a simple Logistic-like equation in which the essential non-linearity is replaced by a feedback loop between the history of activation and an adaptive transition rate that is sensitive to a single dimension of the space of inactive states. This physiologically measurable dimension contributes to the stability of the system and serves as a powerful modulator of input–output relations that depends on the patterns of prior activity; an intrinsic scale free mechanism for cellular adaptation that emerges from the microscopic biophysical properties of ion channels of excitable membranes.

Dissipative stochastic mechanics for capturing neuronal dynamics under the influence of ion channel noise: formalism using a special membrane

Based on the idea conveyed in the author's prior study [Fluct. Noise Lett. 6, L147 (2006)], a physical approach for the description of neuronal dynamics under the influence of ion channel noise is developed in the realm of Nelson's stochastic mechanics when open to dissipative environments. The formalism therein is scrutinized using a special membrane with some tailored properties giving the Rose-Hindmarsh dynamics in the deterministic limit. Led by the presence of multiple number of gates in an ion channel, a dual viewpoint of channel noise is established. Then, stochastic mechanics is adopted to model those channel fluctuations emerging from the uncertainty in accessing the permissible topological states of open gates. A mutual interaction between the above fluctuations and the noise, emerging from the stochasticity in the movement of gating particles between the inner and the outer faces of the membrane, is portrayed within a system plus reservoir strategy. Induced by the interaction, renormalizations of the membrane capacitance and of a membrane voltage dependent potential are found to arise. Consequently, the equations of motion, for the expectation values of the variables and the pair correlation functions, are obtained in the collective membrane voltage space.

Spiking synchronization of ion channel clusters on an axon

Ion channels are distributed in clusters in squid giant axons, rat retinal nerve fiber layers, pyramidal cell dendrites of Apteronotus, etc. Ion channel clusters along the unmyelinated axon generate spontaneous spiking due to ion channel noise. Ion channel clusters are coupled by the axonal cable, and spontaneous spiking of each ion channel cluster can be synchronized. This paper considers the spiking synchronization of two ion channel clusters coupled by an axon. It is shown that axonal parameters affect the spiking synchronization exponentially and ion channel clusters have maximal spiking synchronization when they have the same size. It is further shown that there is an optimal length of the ion channel clusters with maximal spiking synchronization and the optimal length accords with the length of the node of Ranvier in the myelinated axon.

Long-range correlations in rabbit brain neural activity

We have analyzed the presence of persistence properties in rabbit brain electrical signals by means of non-equilibrium statistical physics tools. To measure long-memory properties of these experimental signals, we have first determined whether the data are fractional Gaussian noise (fGn) or fractional Brownian motion (fBm) by calculating the slope of the power spectral density plot of the series. The results show that the series correspond to fBm. Then, the data were studied by means of the bridge detrended scaled windowed variance analysis, detecting long-term correlation. Three different types of experimental signals have been studied: neural basal activity without stimulation, the response induced by a single flash light stimulus and the average of the activity evoked by 200 flash light stimulations. Analysis of the series revealed the existence of persistent behavior in all cases. Moreover, the results also exhibited an increasing correlation in the level of long-term memory from recordings without stimulation, to one sweep recording or 200 sweeps averaged recordings. Thus, brain neural electrical activity is affected not only by its most recent states, but also by previous states much more distant in the past.

Emotional task-dependent low-frequency fluctuations and methylphenidate: Wavelet scaling analysis of 1/f-type fluctuations in fMRI of the cerebellar vermis

Ion channel currents, neural firing patterns, and brain BOLD signals display 1/f-type fluctuations or fractal properties in time. By design, fMRI methods attempt to minimize the contribution of variance from low-frequency physiological 1/f-noise. New fMRI methods are described to visualize and measure 1/f-type BOLD fluctuations in volunteers recalling affectively neutral or emotional memories or meditating (i.e., attending to breathing) then retrospectively rating emotional content. A wavelet scaling exponent (alpha) was used to characterize signals from 0.015625 to 0.5Hz in cerebellar lobules VIII to X of the vermis (posterior inferior vermis; PIV), a region coordinating balance, eye tracking, locomotion, and vascular tone, and a possible site of pathology in attention deficit hyperactivity disorder (ADHD).

RESULTS

Changes in alpha and emotional measures were correlated in PIV voxels (r = 0.622, d.f .= 14, P < 0.0005), but not other regions examined. In contrast, conventional means and standard deviations of PIV voxels were unchanged. Methylphenidate, shown to decrease slow oscillations in rodent basal ganglia [Ruskin DN, Bergstrom DA, Shenker A, Freeman LE, Baek D, Walters JR. Drugs used in the treatment of attention-deficit/hyperactivity disorder affect postsynaptic firing rate and oscillation without preferential dopamine autoreceptor action. Biol Psychiatry 2001;49:340-50.], abolished task-dependent alpha changes in the PIV of an adult with ADHD. Wavelet analysis of long BOLD time series appears well suited to fractal physiology and studies of pharmacologically modulated cerebellar-thalamic-cortical function in ADHD or other psychiatric disorders.

Fractal Stochastic Modeling of Spiking Activity in Suprachiasmatic Nucleus Neurons

Individual neurons in the suprachiasmatic nucleus (SCN), the master biological clock in mammals, autonomously produce highly complex patterns of spikes. We have shown that most (approximately 90%) SCN neurons exhibit truly stochastic interspike interval (ISI) patterns. The aim of this study was to understand the stochastic nature of the firing patterns in SCN neurons by analyzing the ISI sequences of 150 SCN neurons in hypothalamic slices. Fractal analysis, using the periodogram, Fano factor, and Allan factor, revealed the presence of a 1/f-type power-law (fractal) behavior in the ISI sequences. This fractal nature was persistent after the application of the GABAA receptor antagonist bicuculline, suggesting that the fractal stochastic activity is an intrinsic property of individual SCN neurons. Based on these physiological findings, we developed a computational model for the stochastic SCN neurons to find that their stochastic spiking activity was best described by a gamma point process whose mean firing rate was modulated by a fractal binomial noise. Taken together, we suggest that SCN neurons generate temporal spiking patterns using the fractal stochastic point process.

History-dependent multiple-time-scale dynamics in a single-neuron model

History-dependent characteristic time scales in dynamics have been observed at several levels of organization in neural systems. Such dynamics can provide powerful means for computation and memory. At the level of the single neuron, several microscopic mechanisms, including ion channel kinetics, can support multiple-time-scale dynamics. How the temporally complex channel kinetics gives rise to dynamical properties of the neuron is not well understood. Here, we construct a model that captures some features of the connection between these two levels of organization. The model neuron exhibits history-dependent multiple-time-scale dynamics in several effects: first, after stimulation, the recovery time scale is related to the stimulation duration by a power-law scaling; second, temporal patterns of neural activity in response to ongoing stimulation are modulated over time; finally, the characteristic time scale for adaptation after a step change in stimulus depends on the duration of the preceding stimulus. All these effects have been observed experimentally and are not explained by current single-neuron models. The model neuron here presented is composed of an ensemble of ion channels that can wander in a large pool of degenerate inactive states and thus exhibits multiple-time-scale dynamics at the molecular level. Channel inactivation rate depends on recent neural activity, which in turn depends through modulations of the neural response function on the fraction of active channels. This construction produces a model that robustly exhibits nonexponential history-dependent dynamics, in qualitative agreement with experimental results.

Modeling the process of rate selection in neuronal activity

We present the elements of a mathematical computational model that reflects the experimental finding that the time-scale of a neuron is not fixed; but rather varies with the history of its stimulus. Unlike most physiological models, there are no pre-determined rates associated with transitions between states of the system nor are there pre-determined constants associated with adaptation rates; instead, the model is a kind of "modulating automata" where the rates emerge from the history of the system itself. We focus in this paper on the temporal dynamics of a neuron and show how a simple internal structure will give rise to complex temporal behavior. The internal structure modeled here is an abstraction of a reasonably well-understood physiological structure. We also suggest that this behavior can be used to transform a "rate" code into a "temporal one".