The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons

fractal and fractional Article The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons Corina S. Drapaca Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA; [email protected] Abstract: Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases such as epilepsy, multiple sclerosis, and Alzheimer’s disease, which have become more prevalent in the latest years. Keywords: anomalous diffusion; fractional calculus; action potentials; fractional Hodgkin–Huxley model 1. Introduction Myelination of neurons is a critical growth process taking place during the first two years of life, which facilitates the fast transmission of information throughout the body. The Citation: Drapaca, C.S. The Impact process involves the wrapping of the neuronal axons by myelin sheaths made of multiple of Anomalous Diffusion on Action layers of glial plasma membrane. The myelinated regions are separated by the nodes of Potentials in Myelinated Neurons. Fractal Fract. 2021, 5, 4. https://doi. Ranvier (Figure1). Some specialized glial cells called oligodendrocytes produce myelin and org/10.3390/fractalfract5010004 create the neuro–glial-controlled clustering of voltage-gated sodium channels at the nodes of Ranvier and the aggregation of fast voltage-gated potassium channels in the myelin Received: 18 November 2020 sheath [1,2]. The existence of some slow potassium channels in the nodes of Ranvier was Accepted: 29 December 2020 reported in [3]. Experiments performed in cell cultures and in vivo reveal that nodal-like Published: 5 January 2021 groups are created just before the myelination starts [4]. This indicates that myelination is highly dependent on timing and neuronal surroundings, especially the amounts of Publisher’s Note: MDPI stays neu- certain chemicals and water. An impediment to the myelination process or damage to tral with regard to jurisdictional clai- the myelin sheath, which may or may not lead to demyelination, could cause serious, ms in published maps and institutio- possibly life-threatening, neurologic disorders. For instance, multiple sclerosis is the most nal affiliations. common demyelinating disorder affecting young adults that currently has no cure [5]. Mathematical models can provide essential insights into myelination/demyelination processes that can further inspire better diagnostic and therapeutic procedures for various neurological disorders. Copyright: © 2021 by the author. Li- censee MDPI, Basel, Switzerland. Neurons transmit information throughout the body via action potentials. In a myeli- This article is an open access article nated neuron an action potential happens when sodium flows inside the neuron at the distributed under the terms and con- node of Ranvier, and potassium is released into the extracellular space between the axon ditions of the Creative Commons At- and the myelin (the periaxonal space). To prevent the accumulation of potassium and tribution (CC BY) license (https:// osmotically driven water in the periaxonal space, potassium diffuses through the myelin’s creativecommons.org/licenses/by/ potassium channels and the gap junctions connecting the myelin layers and is removed by 4.0/). the astrocyte endfeet that form gap junctions with the outermost myelin layer [6]. Thus, in Fractal Fract. 2021, 5, 4. https://doi.org/10.3390/fractalfract5010004 https://www.mdpi.com/journal/fractalfract Fractal Fract. 2021, 5, x FOR PEER REVIEW 3 of 20 Fractal Fract. 2021, 5, 4 2 of 19 The structure of the paper is as follows. The mathematical preliminaries needed in the paper are given in Section 2. The derivation of the spatio-temporal fractional cable equationmyelinated is presented neurons, thein Section action potentials3, together do with not a propagatenumerical withinsolution the to axonsthe equation but happen cou- pledonly with at the the nodes fractional of Ranvier Hodgkin–Huxley (Figure1). Theequa worktions. hypothesis Numerical ofsimulations this paper are is shown that the in Sectionanomalous 4. The diffusion paper ends of ions with through a section the of neuronal conclusions membrane and further at the work. nodes of Ranvier and outside the neurons can influence the propagation of action potentials. Figure 1. InIn a a myelinated myelinated neuron, neuron, action action potentials (blue(blue verticalvertical doubledouble arrows)arrows) happenhappen onlyonly at at the thenodes nodes of Ranvier of Ranvier andpropagate and propagate between between the cell the body ce andll body the axonand the terminals axon te (bidirectionalrminals (bidirectional propagation) . propagation). It was observed experimentally that the voltage-gated potassium channel Kv2.1 in 2.the Mathematical plasma membrane Preliminaries of a cultured human embryotic kidney cell displays anomalous diffusion [7]. Because the Kv2.1 channel shows similar clustering patterns in the membranes The fractional calculus results needed in the paper are given in this section [27–32]. of cultured human embryonic kidney cells and native neurons [7], it is possible that anomalous diffusion of potassium can happen in the myelin sheath of neurons as well. Definition 1: The crowdedness of the very narrow ion channels [8] could be the cause of the anomalous 1.diffusion [28] (p. of ions33); through[29]: Let the∈ membrane’s(, ) and channels. >0. The Anomalous left-sided Riemann–Liouville diffusion of ions that fractional could affectintegral the propagation of order ofis action potentials does not happen only through the ion channels of the neuronal membrane. According to [4], clusters of cytoskeletal scaffolding proteins, cell-adhesion molecules, and parts of the extracellular1 () matrix (ECM) can be found near the () = (1) nodes of Ranvier, which could influence() ion flow.( Furthermore, − ) anomalous diffusion of ions through the extracellular space (ECS) is also possible due to (1) the ECS geometry; and(2) deadthe right-sided spaces of Riemann–Liouville the ECS, voids, or fractional glial wrapping integral aroundof order cells; is (3) flow obstruction by the ECM; (4) binding sites on the cell membranes of the ECM; (5) the presence of fixed negative charges on the ECM [9] that control1 ion mobility() [10]; and (6) the tight control () = (2) of the ion and water movement in the ECS() by astrocytes( − ) [6,11]. In particular, anomalous diffusion might be involved in the regulation of the extracellular potassium concentration by astrocytes, which is essential for the proper creation and propagation of action potentials where () = is the gamma function. In particular, () =() =(). (for example, an unregulated elevated extracellular potassium concentration can cause an () 2.epileptic [27,29,31]: seizure If [12:]). [, Thus, ] anomalous→ is -times diffusion differentiable of ions on and(, water ) ⊂in with the vicinity ∈ of(, the ) nodesand of Ranvier − 1 < is caused < , by a combination=, 1,2,3… then the left-sided of structural Caputo geometry fractional and derivative highly of dynamic order and possiblyis emerging biochemical processes. Anomalous diffusion through various materials has been successfully modeled using 1 ()() fractional calculus [13]. A few generalizations of the classic cable equation() that models () = = () (3) the spatio-temporal propagation( of action − ) ( potentials − ) in neurons that involve fractional derivatives have been proposed in the literature. In [14,15], a fractional cable equation andwas the proposed right-sided that Caputo uses fractional a temporal derivative Riemann–Liouville of order is fractional derivative acting on the classic Laplace operator to model anomalous electro-diffusion in neurons. In [16], () a space-fractional cable equation(−1) was derived () to model the non-local forward (from the () = = (−1)()() (4) neuronal body to the axon terminals)( − ) propagation( − ) of action potentials in mature myelinated neurons. In this paper, a spatio-temporal fractional cable equation is proposed that involves () () whereboth spatial is andthe temporalth-order derivative fractional of derivatives. In particular, The equation() = models() the, and bidirectional () = ((forward−1)() and(). backward) propagation of the action potentials in the presence of anomalous 3.diffusion. [27,30,31]: The proposedThe left (right)-sided work is a Caputo straightforward fractional generalizationderivative of order of the,mathematical with −1< model<,=2,3,… in [16]. In the case, is said when to be the a left(right)-sided order of thetemporal sequential derivativeCaputo fractional is 1 and derivative only the if forward propagation

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