Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions

A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation

In this article, we developed a new higher-order implicit finite difference iterative scheme (FDIS) for the solution of the two dimension (2-D) time fractional Cable equation (FCE). In the new proposed FDIS, the time fractional and space derivatives are discretized using the Caputo fractional derivative and fourth-order implicit scheme, respectively. Moreover, the proposed scheme theoretical analysis (convergence and stability) is also discussed using the Fourier analysis method. Finally, some numerical test problems are presented to show the effectiveness of the proposed method.

An Efficient Technique to Solve Time-Fractional Kawahara and Modified Kawahara Equations

In this article, we analysed the approximate solutions of the time-fractional Kawahara equation and modified Kawahara equation, which describe the propagation of signals in transmission lines and the formation of nonlinear water waves in the long wavelength region. An efficient technique, namely the natural transform decomposition method, is used in the present study. Fractional derivatives are considered in Caputo, Caputo–Fabrizio, and Atangana–Baleanu operative in the Caputo manner. We have presented numerical results graphically to demonstrate the applicability and efficiency of derivatives with fractional order to depict the water waves in long wavelength regions. The symmetry pattern is a fundamental feature of the Kawahara equation and the symmetrical aspect of the solution can be seen from the graphical representations. The obtained outcomes of the proposed method are compared to those of other well-known numerical techniques, such as the homotopy analysis method and residual power series method. Numerical solutions converge to the exact solution of the Kawahara equations, demonstrating the significance of our proposed method.

Fractional-Order Traveling Wave Approximations for a Fractional-Order Neural Field Model

In this work, we establish a fractional-order neural field mathematical model with Caputo's fractional derivative temporal order α considering 0 < α < 2, to analyze the effect of fractional-order on cortical wave features observed preceding seizure termination. The importance of this incorporation relies on the theoretical framework established by fractional-order derivatives in which memory and hereditary properties of a system are considered. Employing Mittag-Leffler functions, we first obtain approximate fractional-order solutions that provide information about the initial wave dynamics in a fractional-order frame. We then consider the Adomian decomposition method to approximate pulse solutions in a wider range of orders and longer times. The former approach establishes a direct way to investigate the initial relationships between fractional-order and wave features, such as wave speed and wave width. In contrast, the latter approach displays wave propagation dynamics in different fractional orders for longer times. Using the previous two approaches, we establish approximate wave solutions with characteristics consistent with in vivo cortical waves preceding seizure termination. In our analysis, we find consistent differences in the initial effect of the fractional-order on the features of wave speed and wave width, depending on whether α <1 or α>1. Both cases can model the shape of cortical wave propagation for different fractional-orders at the cost of modifying the wave speed. Our results also show that the effect of fractional-order on wave width depends on the synaptic threshold and the synaptic connectivity extent. Fractional-order derivatives have been interpreted as the memory trace of the system. This property and the results of our analysis suggest that fractional-order derivatives and neuronal collective memory modify cortical wave features.

Increased Signal Delays and Unaltered Synaptic Input Pattern Recognition in Layer III Neocortical Pyramidal Neurons of the rTg4510 Mouse Model of Tauopathy: A Computer Simulation Study With Passive Membrane

Abnormal tau proteins are involved in pathology of many neurodegenerative disorders. Transgenic rTg4510 mice express high levels of human tau protein with P301L mutation linked to chromosome 17 that has been associated with frontotemporal dementia with parkinsonism. By 9 months of age, these mice recapitulate key features of human tauopathies, including presence of hyperphosphorylated tau and neurofibrillary tangles (NFTs) in brain tissue, atrophy and loss of neurons and synapses, and hyperexcitability of neurons, as well as cognitive deficiencies. We investigated effects of such human mutant tau protein on neuronal membrane, subthreshold dendritic signaling, and synaptic input pattern recognition/discrimination in layer III frontal transgenic (TG) pyramidal neurons of 9-month-old rTg4510 mice and compared these characteristics to those of wild-type (WT) pyramidal neurons from age-matched control mice. Passive segmental cable models of WT and TG neurons were set up in the NEURON simulator by using three-dimensionally reconstructed morphology and electrophysiological data of these cells. Our computer simulations predict leakage resistance and capacitance of neuronal membrane to be unaffected by the mutant tau protein. Computer models of TG neurons showed only modest alterations in distance dependence of somatopetal voltage and current transfers along dendrites and in rise times and half-widths of somatic Excitatory Postsynaptic Potential (EPSPs) relative to WT control. In contrast, a consistent and statistically significant slowdown was detected in the speed of simulated subthreshold dendritic signal propagation in all regions of the dendritic surface of mutant neurons. Predictors of synaptic input pattern recognition/discrimination remained unaltered in model TG neurons. This suggests that tau pathology is primarily associated with failures/loss in synaptic connections rather than with altered intraneuronal synaptic integration in neurons of affected networks.

Fractional cable equation for general geometry: A model of axons with swellings and anomalous diffusion

Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.

Electrodiffusion phenomena in neuroscience: a neglected companion

The emerging technological revolution in genetically encoded molecular sensors and super-resolution imaging provides neuroscientists with a pass to the real-time nano-world. On this small scale, however, classical principles of electrophysiology do not always apply. This is in large part because the nanoscopic heterogeneities in ionic concentrations and the local electric fields associated with individual ions and their movement can no longer be ignored. Here, we review basic principles of molecular electrodiffusion in the cellular environment of organized brain tissue. We argue that accurate interpretation of physiological observations on the nanoscale requires a better understanding of the underlying electrodiffusion phenomena.

Induced mitochondrial membrane potential for modeling solitonic conduction of electrotonic signals

A cable model that includes polarization-induced capacitive current is derived for modeling the solitonic conduction of electrotonic potentials in neuronal branchlets with microstructure containing endoplasmic membranes. A solution of the nonlinear cable equation modified for fissured intracellular medium with a source term representing charge ‘soakage’ is used to show how intracellular capacitive effects of bound electrical charges within mitochondrial membranes can influence electrotonic signals expressed as solitary waves. The elastic collision resulting from a head-on collision of two solitary waves results in localized and non-dispersing electrical solitons created by the nonlinearity of the source term. It has been shown that solitons in neurons with mitochondrial membrane and quasi-electrostatic interactions of charges held by the microstructure (i.e., charge ‘soakage’) have a slower velocity of propagation compared with solitons in neurons with microstructure, but without endoplasmic membranes. When the equilibrium potential is a small deviation from rest, the nonohmic conductance acts as a leaky channel and the solitons are small compared when the equilibrium potential is large and the outer mitochondrial membrane acts as an amplifier, boosting the amplitude of the endogenously generated solitons. These findings demonstrate a functional role of quasi-electrostatic interactions of bound electrical charges held by microstructure for sustaining solitons with robust self-regulation in their amplitude through changes in the mitochondrial membrane equilibrium potential. The implication of our results indicate that a phenomenological description of ionic current can be successfully modeled with displacement current in Maxwell’s equations as a conduction process involving quasi-electrostatic interactions without the inclusion of diffusive current. This is the first study in which solitonic conduction of electrotonic potentials are generated by polarization-induced capacitive current in microstructure and nonohmic mitochondrial membrane current.

Distributed-order diffusion equations and multifractality: Models and solutions

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

Numerical simulation of fractional Cable equation of spiny neuronal dendrites

In this article, numerical study for the fractional Cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. A simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method.

Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model

The voltage trace of neuronal activities can follow multiple timescale dynamics that arise from correlated membrane conductances. Such processes can result in power-law behavior in which the membrane voltage cannot be characterized with a single time constant. The emergent effect of these membrane correlations is a non-Markovian process that can be modeled with a fractional derivative. A fractional derivative is a non-local process in which the value of the variable is determined by integrating a temporal weighted voltage trace, also called the memory trace. Here we developed and analyzed a fractional leaky integrate-and-fire model in which the exponent of the fractional derivative can vary from 0 to 1, with 1 representing the normal derivative. As the exponent of the fractional derivative decreases, the weights of the voltage trace increase. Thus, the value of the voltage is increasingly correlated with the trajectory of the voltage in the past. By varying only the fractional exponent, our model can reproduce upward and downward spike adaptations found experimentally in neocortical pyramidal cells and tectal neurons in vitro. The model also produces spikes with longer first-spike latency and high inter-spike variability with power-law distribution. We further analyze spike adaptation and the responses to noisy and oscillatory input. The fractional model generates reliable spike patterns in response to noisy input. Overall, the spiking activity of the fractional leaky integrate-and-fire model deviates from the spiking activity of the Markovian model and reflects the temporal accumulated intrinsic membrane dynamics that affect the response of the neuron to external stimulation.

Spike adaptation is a property of most neurons. When spike time adaptation occurs over multiple time scales, the dynamics can be described by a power-law. We study the computational properties of a leaky integrate-and-fire model with power-law adaptation. Instead of explicitly modeling the adaptation process by the contribution of slowly changing conductances, we use a fractional temporal derivative framework. The exponent of the fractional derivative represents the degree of adaptation of the membrane voltage, where 1 is the normal leaky integrator while values less than 1 produce increasing correlations in the voltage trace. The temporal correlation is interpreted as a memory trace that depends on the value of the fractional derivative. We identify the memory trace in the fractional model as the sum of the instantaneous differentiation weighted by a function that depends on the fractional exponent, and it provides non-local information to the incoming stimulus. The spiking dynamics of the fractional leaky integrate-and-fire model show memory dependence that can result in downward or upward spike adaptation. Our model provides a framework for understanding how long-range membrane voltage correlations affect spiking dynamics and information integration in neurons.

Electrodiffusive model for astrocytic and neuronal ion concentration dynamics

The cable equation is a proper framework for modeling electrical neural signalling that takes place at a timescale at which the ionic concentrations vary little. However, in neural tissue there are also key dynamic processes that occur at longer timescales. For example, endured periods of intense neural signaling may cause the local extracellular K⁺-concentration to increase by several millimolars. The clearance of this excess K⁺ depends partly on diffusion in the extracellular space, partly on local uptake by astrocytes, and partly on intracellular transport (spatial buffering) within astrocytes. These processes, that take place at the time scale of seconds, demand a mathematical description able to account for the spatiotemporal variations in ion concentrations as well as the subsequent effects of these variations on the membrane potential. Here, we present a general electrodiffusive formalism for modeling of ion concentration dynamics in a one-dimensional geometry, including both the intra- and extracellular domains. Based on the Nernst-Planck equations, this formalism ensures that the membrane potential and ion concentrations are in consistency, it ensures global particle/charge conservation and it accounts for diffusion and concentration dependent variations in resistivity. We apply the formalism to a model of astrocytes exchanging ions with the extracellular space. The simulations show that K⁺-removal from high-concentration regions is driven by a local depolarization of the astrocyte membrane, which concertedly (i) increases the local astrocytic uptake of K⁺, (ii) suppresses extracellular transport of K⁺, (iii) increases axial transport of K⁺ within astrocytes, and (iv) facilitates astrocytic relase of K⁺ in regions where the extracellular concentration is low. Together, these mechanisms seem to provide a robust regulatory scheme for shielding the extracellular space from excess K⁺.

When neurons generate electrical signals they release potassium ions (K⁺) into the extracellular space. During periods of intense neural activity, the local extracellular K⁺ may increase drastically. If it becomes too high, it can lead to neural dysfunction. Astrocytes (a kind of glial cells) are involved in preventing this from happening. Astrocytes can take up excess K⁺, transport it intracellularly, and release it in regions where the concentration is lower. This process is called spatial buffering, and a full mechanistic understanding of it is currently lacking. The aim of this work is twofold: First, we develop a formalism for modeling ion concentration dynamics in the intra- and extracellular space. The formalism is general, and could be used to simulate many cellular processes. It accounts for ion transports due to diffusion (along concentration gradients) as well as electrical migration (along voltage gradients). It extends previous, related formalisms, which have focused only on intracellular dynamics. Secondly, we apply the formalism to model how astrocytes exchange ions with the extracellular space. We conclude that the membrane mechanisms possessed by astrocytes seem optimal for shielding the extracellular space from excess K⁺, and provide a full mechanistic description of the spatial (K⁺) buffering process.

Intracellular capacitive effects of polarized proteins in dendrites

Passive dendrites become active as a result of electrostatic interactions by dielectric polarization in proteins in a segment of a dendrite. The resultant nonlinear cable equation for a cylindrical volume representation of a dendritic segment is derived from Maxwell's equations under assumptions: (i) the electric field is restricted longitudinally along the cable length; (ii) extracellular isopotentiality; (iii) quasi-electrostatic conditions; (iv) isotropic membrane and homogeneous medium with constant conductivity; and (v) protein polarization contributes to intracellular capacitive effects through a well defined nonlinear capacity-voltage characteristic; (vi) intracellular resistance and capacitance in parallel are connected to the membrane in series. Under the above hypotheses, traveling wave solutions of the cable equation are obtained as propagating fronts of electrical excitation associated with capacitive charge-equalization and dispersion of continuous polarization charge densities in an Ohmic cable. The intracellular capacitative effects of polarized proteins in dendrites contribute to the conduction process.

Moving boundary problems governed by anomalous diffusion

Anomalous diffusion can be characterized by a mean-squared displacement 〈x(2)(t)〉 that is proportional to t(α) where α≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (α<1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.

Anomalous transport and nonlinear reactions in spiny dendrites

We present a mesoscopic description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in Fedotov [Phys. Rev. Lett. 101, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one cannot separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particle's position <x(t)> increases as t(μ) with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

Fractional chemotaxis diffusion equations

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.

Dendritic vulnerability in neurodegenerative disease: insights from analyses of cortical pyramidal neurons in transgenic mouse models

In neurodegenerative disorders, such as Alzheimer's disease, neuronal dendrites and dendritic spines undergo significant pathological changes. Because of the determinant role of these highly dynamic structures in signaling by individual neurons and ultimately in the functionality of neuronal networks that mediate cognitive functions, a detailed understanding of these changes is of paramount importance. Mutant murine models, such as the Tg2576 APP mutant mouse and the rTg4510 tau mutant mouse have been developed to provide insight into pathogenesis involving the abnormal production and aggregation of amyloid and tau proteins, because of the key role that these proteins play in neurodegenerative disease. This review showcases the multidimensional approach taken by our collaborative group to increase understanding of pathological mechanisms in neurodegenerative disease using these mouse models. This approach includes analyses of empirical 3D morphological and electrophysiological data acquired from frontal cortical pyramidal neurons using confocal laser scanning microscopy and whole-cell patch-clamp recording techniques, combined with computational modeling methodologies. These collaborative studies are designed to shed insight on the repercussions of dystrophic changes in neocortical neurons, define the cellular phenotype of differential neuronal vulnerability in relevant models of neurodegenerative disease, and provide a basis upon which to develop meaningful therapeutic strategies aimed at preventing, reversing, or compensating for neurodegenerative changes in dementia.

Thermal noise due to surface-charge effects within the Debye layer of endogenous structures in dendrites

An assumption commonly used in cable theory is revised by taking into account electrical amplification due to intracellular capacitive effects in passive dendritic cables. A generalized cable equation for a cylindrical volume representation of a dendritic segment is derived from Maxwell's equations under assumptions: (i) the electric-field polarization is restricted longitudinally along the cable length; (ii) extracellular isopotentiality; (iii) quasielectrostatic conditions; and (iv) homogeneous medium with constant conductivity and permittivity. The generalized cable equation is identical to Barenblatt's equation arising in the theory of infiltration in fissured strata with a known analytical solution expressed in terms of a definite integral involving a modified Bessel function and the solution to a linear one-dimensional classical cable equation. Its solution is used to determine the impact of thermal noise on voltage attenuation with distance at any particular time. A regular perturbation expansion for the membrane potential about the linear one-dimensional classical cable equation solution is derived in terms of a Green's function in order to describe the dynamics of free charge within the Debye layer of endogenous structures in passive dendritic cables. The asymptotic value of the first perturbative term is explicitly evaluated for small values of time to predict how the slowly fluctuating (in submillisecond range) electric field attributed to intracellular capacitive effects alters the amplitude of the membrane potential. It was found that capacitive effects are almost negligible for cables with electrotonic lengths L>0.5 , contributes up to 10% of the signal for cables with electrotonic lengths in the range between 0.25<L<0.5 , and dominates the membrane potential for electrotonically short cables (L<0.2) . These results show that electrotonically short dendritic cables with both ends sealed are prone to significant neurobiological thermal noise due to intracellular capacitive effects. The presence of significant thermal noise weakens the assumption of intracellular isopotentiality when approximating dendrites with compartments.