Membrane Capacitive Memory Alters Spiking in Neurons Described by the Fractional-Order Hodgkin-Huxley Model
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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.
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“This similar methodology, both model interactions and parameter values, has been used in recent studies61–63.”
From Perineuronal nets decrease membrane capacitance of peritumoral fast spiking interneurons in a model of epilepsy (2018) · cites this paper as [63]
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Cites
Hodgkin, A. L., & Huxley, A. F. (1952) · The Journal of Physiology · doi:10.1113/jphysiol.1952.sp004764 · Cited by 23,191 · see citing works
Fractional Differential Equations
Podlubný, I. (2025) · MDPI eBooks · doi:10.3390/books978-3-7258-4742-6 · Cited by 20,452 · see citing works
(1999) · Mathematics in Science and Engineering/Mathematics in science and engineering · doi:10.1016/s0076-5392(99)x8001-5 · Cited by 5,009 · see citing works
Ion Channels/Excitable Membranes
(2011) · Springer eBooks · doi:10.1007/springerreference_33998 · Cited by 4,066 · see citing works
Cellular basis of working memory
Goldman-Rakic, P. S. (1995) · Neuron · doi:10.1016/0896-6273(95)90304-6 · Cited by 2,663 · see citing works
Koch, C. (1998) · Oxford University Press eBooks · doi:10.1093/oso/9780195104912.001.0001 · Cited by 2,306 · see citing works
Neuron Activity Related to Short-Term Memory
Fuster, J. M., & Alexander, G. E. (1971) · Science · doi:10.1126/science.173.3997.652 · Cited by 2,085 · see citing works
Fractional Calculus in Bioengineering
West, B. J. (2007) · Journal of Statistical Physics · doi:10.1007/s10955-007-9294-0 · Cited by 1,996 · see citing works
Finite difference/spectral approximations for the time-fractional diffusion equation
Lin, Y., & Xu, C. (2007) · Journal of Computational Physics · doi:10.1016/j.jcp.2007.02.001 · Cited by 1,848 · see citing works
Ion Channels/Excitable Membranes
Terlau, H., & Kirchhoff, F. (2006) · Springer eBooks · doi:10.1007/3-540-29623-9_5640 · Cited by 1,805 · see citing works
Fractional calculus models of complex dynamics in biological tissues
Magin, R. L. (2009) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2009.08.039 · Cited by 1,019 · see citing works
Fractional Dynamics and Control
Bǎleanu, D., Machado, J., & Luo, A. (2011) · Springer eBooks · doi:10.1007/978-1-4614-0457-6 · Cited by 812 · see citing works
Westerlund, S., & Ekstam, L. (1994) · IEEE Transactions on Dielectrics and Electrical Insulation · doi:10.1109/94.326654 · Cited by 787 · see citing works
McIntyre, C. C., Richardson, A. G., & Grill, W. M. (2002) · Journal of Neurophysiology · doi:10.1152/jn.00353.2001 · Cited by 764 · see citing works
Fractional differentiation by neocortical pyramidal neurons
Lundstrom, B. N., Higgs, M. H., Spain, W. J., & Fairhall, A. L. (2008) · Nature Neuroscience · doi:10.1038/nn.2212 · Cited by 677 · see citing works
Fractional Calculus in Bioengineering, Part 1
Magin, R. L. (2004) · Critical Reviews in Biomedical Engineering · doi:10.1615/critrevbiomedeng.v32.10 · Cited by 621 · see citing works
Alternating and Synchronous Rhythms in Reciprocally Inhibitory Model Neurons
Wang, X., & Rinzel, J. (1992) · Neural Computation · doi:10.1162/neco.1992.4.1.84 · Cited by 607 · see citing works
Recherches quantitatives sur l' excitation electrique des nerfs traitee comme une polarization.
Lapique, L. (1907) · Medical Entomology and Zoology · Cited by 557 · see citing works
The Grünwald–Letnikov method for fractional differential equations
Scherer, R., Kalla, S. L., Tang, Y., et al. (2011) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2011.03.054 · Cited by 482 · see citing works
Numerical comparison of methods for solving linear differential equations of fractional order
Momani, S., & Odibat, Z. (2005) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2005.10.068 · Cited by 394 · see citing works
On the stability of linear systems with fractional-order elements
Radwan, A. G., Soliman, A. M., Elwakil, A. S., et al. (2007) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2007.10.033 · Cited by 329 · see citing works
The fractional-order dynamics of brainstem vestibulo-oculomotor neurons
Anastasio, T. J. (1994) · Biological Cybernetics · doi:10.1007/bf00206239 · Cited by 314 · see citing works
On the origin of capacitance dispersion of rough electrodes
Kerner, Z., & Pajkossy, T. (2000) · Electrochimica Acta · doi:10.1016/s0013-4686(00)00574-0 · Cited by 301 · see citing works
Calcium Coding and Adaptive Temporal Computation in Cortical Pyramidal Neurons
Wang, X. (1998) · Journal of Neurophysiology · doi:10.1152/jn.1998.79.3.1549 · Cited by 285 · see citing works
Sur la possibilite de rendre comparables entre eux les appareils servant a l'excitation electrique.
Weiss, G. (1990) · ARCHIVES ITALIENNES DE BIOLOGIE · doi:10.4449/aib.v35i1.1355 · Cited by 282 · see citing works
Fractional Calculus in Bioengineering, Part 1
Magin, R. L. (2004) · Critical Reviews in Biomedical Engineering · doi:10.1615/critrevbiomedeng.v32.i1.10 · Cited by 246 · see citing works
ELECTRIC CONDUCTANCE OF BIOLOGICAL SYSTEMSPreprint
Cole, K. S. (1933) · Cold Spring Harbor Symposia on Quantitative Biology · doi:10.1101/sqb.1933.001.01.014 · Cited by 225 · see citing works
The Stochastic Evolution of a Protocell: The Gillespie Algorithm in a Dynamically Varying Volume
Carletti, T., & Filisetti, A. (2012) · Computational and Mathematical Methods in Medicine · doi:10.1155/2012/423627 · Cited by 207 · see citing works
Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus
Magin, R. L., & Ovadia, M. (2008) · Journal of Vibration and Control · doi:10.1177/1077546307087439 · Cited by 205 · see citing works
Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations
Rinzel, J., & Miller, R. N. (1980) · Mathematical Biosciences · doi:10.1016/0025-5564(80)90109-1 · Cited by 205 · see citing works
Fractional Calculus in Bioengineering, Part3
Magin, R. L. (2004) · Critical Reviews in Biomedical Engineering · doi:10.1615/critrevbiomedeng.v32.i34.10 · Cited by 185 · see citing works
Ion channel kinetics: a model based on fractal scaling rather than multistate Markov processes
Liebovitch, L. S., Fischbarg, J., & Koniarek, J. P. (1987) · Mathematical Biosciences · doi:10.1016/0025-5564(87)90042-3 · Cited by 179 · see citing works
ELECTRIC PHASE ANGLE OF CELL MEMBRANES
Cole, K. S. (1932) · The Journal of General Physiology · doi:10.1085/jgp.15.6.641 · Cited by 170 · see citing works
Fractional kinetics in drug absorption and disposition processes
Dokoumetzidis, A., & Macheras, P. E. (2009) · Journal of Pharmacokinetics and Pharmacodynamics · doi:10.1007/s10928-009-9116-x · Cited by 152 · see citing works
Zur Theorie des elektrischen Reizes
Nernst, W. (1908) · Pflügers Archiv - European Journal of Physiology · doi:10.1007/bf01677956 · Cited by 142 · see citing works
Neuronal Spike Timing Adaptation Described with a Fractional Leaky Integrate-and-Fire Model
Teka, W. W., Marinov, T. M., & Santamarı́a, F. (2014) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1003526 · Cited by 140 · see citing works
The fractional-order modeling and synchronization of electrically coupled neuron systems
Moaddy, K., Radwan, A. G., Salama, K. N., et al. (2012) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2012.01.005 · Cited by 136 · see citing works
Fractional kinetics in multi-compartmental systems
Dokoumetzidis, A., Magin, R. L., & Macheras, P. E. (2010) · Journal of Pharmacokinetics and Pharmacodynamics · doi:10.1007/s10928-010-9170-4 · Cited by 125 · see citing works
Fractional diffusion modeling of ion channel gating
Goychuk, I., & Peter Hänggi, A. (2004) · Physical Review E · doi:10.1103/physreve.70.051915 · Cited by 120 · see citing works
Simulation of drug uptake in a two compartmental fractional model for a biological system
Petráš, I., & Magin, R. L. (2011) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2011.02.012 · Cited by 118 · see citing works
Fractal ion-channel behavior generates fractal firing patterns in neuronal models
Lowen, S. B., Liebovitch, L. S., & White, J. A. (1999) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.59.5970 · Cited by 96 · see citing works
Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model
Dong, J., Zhang, G.-J., Xie, Y., et al. (2013) · Cognitive Neurodynamics · doi:10.1007/s11571-013-9273-x · Cited by 87 · see citing works
Abundant bursting patterns of a fractional-order Morris–Lecar neuron model
Shi, M., & Wang, Z. (2013) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2013.10.032 · Cited by 75 · see citing works
An anomalous subdiffusion model for calcium spark in cardiac myocytes
Tan, W., Fu, C., Fu, C., et al. (2007) · Applied Physics Letters · doi:10.1063/1.2805208 · Cited by 74 · see citing works
ELECTRIC IMPEDANCE OF NERVE AND MUSCLEPreprint
Cole, K. S., & Curtis, H. J. (1936) · Cold Spring Harbor Symposia on Quantitative Biology · doi:10.1101/sqb.1936.004.01.010 · Cited by 72 · see citing works
An efficient method for solving fractional Hodgkin–Huxley model
Nagy, A. M., & Sweilam, N. H. (2014) · Physics Letters A · doi:10.1016/j.physleta.2014.06.012 · Cited by 67 · see citing works
Comparison of numerical methods for fractional differential equations
Connolly, J. A., & Ford, N. J. (2006) · Communications on Pure & Applied Analysis · doi:10.3934/cpaa.2006.5.289 · Cited by 63 · see citing works
Parallelising the Computation of Minimal Absent Words
Barton, C., Héliou, A., Mouchard, L., et al. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-32152-3_23 · Cited by 57 · see citing works
Siwy, Z. S., Ausloos, M., & Ivanova, K. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.65.031907 · Cited by 50 · see citing works
Hurst Analysis Applied to the Study of Single Calcium-activated Potassium Channel Kinetics
Varanda, W. A., Liebovitch, L. S., Figueirôa, J. N., & Nogueira, R. A. (2000) · Journal of Theoretical Biology · doi:10.1006/jtbi.2000.2131 · Cited by 50 · see citing works
Fractional dynamics from the ordinary Langevin equation
Stanislavsky, A. A. (2003) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.67.021111 · Cited by 49 · see citing works
Alternating Current Conductance and Direct Current Excitation of Nerve
Cole, K. S. (1934) · Science · doi:10.1126/science.79.2042.164 · Cited by 38 · see citing works
Phase-dependent stimulation effects on bursting activity in a neural network cortical simulation
Anderson, W. S., Kudela, P., Weinberg, S. H., Bergey, G. K., & Franaszczuk, P. J. (2009) · Epilepsy Research · doi:10.1016/j.eplepsyres.2008.12.005 · Cited by 34 · see citing works
Self-citationParallel Processing and Applied Mathematics
MV, A., D’Ambrosio, D., V, L., Rongo, R., & S., S. (2010) · Lecture notes in computer science · doi:10.1007/978-3-642-14403-5 · Cited by 30 · see citing works
Weinberg, S. H., & Smith, G. D. (2014) · Biophysical Journal · doi:10.1016/j.bpj.2014.04.045 · Cited by 29 · see citing works
Self-citationYao, K., Liang, Y., & Zhang, F. (2008) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2008.09.053 · Cited by 26 · see citing works
Chaotic model and memory in single calcium-activated potassium channel kinetics
Bandeira, H. T., Barbosa, C. T. F., de Oliveira, R. A. C., et al. (2008) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.2944980 · Cited by 23 · see citing works
Using Fractional Derivatives to Generalize the Hodgkin–Huxley Model
Sherief, H. H., El-Sayed, A. M. A., Behiry, S. H., et al. (2011) · Springer eBooks · doi:10.1007/978-1-4614-0457-6_23 · Cited by 21 · see citing works
High-Frequency Stimulation of Excitable Cells and Networks
Weinberg, S. H. (2013) · PLoS ONE · doi:10.1371/journal.pone.0081402 · Cited by 21 · see citing works
Self-citationParallel Processing and Applied Mathematics
Wyrzykowski, R., WaÅniewski, J., Paprzycki, M., et al. (2002) · Lecture notes in computer science · doi:10.1007/3-540-48086-2 · Cited by 17 · see citing works
Weinberg, S. H., & Smith, G. D. (2012) · Computational and Mathematical Methods in Medicine · doi:10.1155/2012/897371 · Cited by 16 · see citing works
Self-citationRepresentation of Collective Electrical Behavior of Cardiac Cell Sheets
Weinberg, S. H., Iravanian, S., & Tung, L. (2008) · Biophysical Journal · doi:10.1529/biophysj.107.128207 · Cited by 13 · see citing works
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Tewari, B. P., Chaunsali, L., Campbell, S. L., Patel, D. C., Goode, A. E., & Sontheimer, H. (2018) · Nature Communications · doi:10.1038/s41467-018-07113-0 · Cited by 208 · see citing works
- Authors
- Weinberg, S. H.
- Affiliations
- Old Dominion University (US)
- Corresponding author
- Weinberg, Seth H.
- ORCID
- Weinberg, Seth H.
- Venue
- PLOS ONE
- Publisher
- Public Library of Science (PLoS)
- Published
- 2015-05-13
- Volume
- 10
- Issue
- 5
- Pages
- e0126629
- ISSN
- 1932-6203 (ISSN-L)
- PMC
- PMC4430543
- OpenAlex
- W226832100
- Cited by
- 76
- References
- 62
- Citations by year
- 2026: 10 · 2025: 7 · 2024: 5 · 2023: 7 · 2022: 9 · 2021: 5 · 2020: 8 · 2019: 8 · 2018: 5 · 2017: 9 · 2016: 3
- FWCI
- 5.72
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- PLoS ONE (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Figshare (submittedVersion, cc-by)
- Funders
- Old Dominion University
- Topics
- Physical Sciences › Mathematics › Modeling and Simulation › Fractional Differential Equations Solutions
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › stochastic dynamics and bifurcation
- Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
- Keywords
- fractional derivatives, neuronal spiking, membrane potential dynamics, sodium and potassium currents, axons, neural network dynamics
- MeSH terms
- Action Potentials, Algorithms, Axons, Models, Neurological, Neurons
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