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Low-dimensional models of single neurons: a review

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Abstract

The classical Hodgkin–Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as subthreshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type.

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Acknowledgements

The authors acknowledge support from the NSF Grants CRCNS-DMS-1608077 (HGR) and IOS-2002863 (HGR), from CONICET, Argentina (UC), and the Fulbright Program (VGB). The authors are thankful to John Rinzel for useful suggestions and discussions. This paper benefited from discussions held as part of the workshop “Current and Future Theoretical Frameworks in Neuroscience” (San Antonio, TX, Feb 4–8, 2019) supported by the NSF Grants DBI-1820631 (HGR) and IOS-1516648 (Fidel Santamaría, co-organizer). This paper also benefited from discussions during the course on “Reduced and simplified spiking neuron models” taught at the VIII Latin American School on Computational Neuroscience (LASCON 2020) organized by Antonio Roque (USP, Brazil) and supported by FAPESP Grants 2013/07699-0 (NeuroMat) and 2019/10496-0 and the IBRO-LARC Schools Funding Program. The authors are grateful to an anonymous reviewer for useful comments and suggestions.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Nacional del Sur and CONICET, Bahía Blanca, Buenos Aires, Argentina

    Ulises Chialva

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Vicente González Boscá

  3. Federated Department of Biological Sciences, New Jersey Institute of Technology and Rutgers University, Newark, New Jersey, USA

    Horacio G. Rotstein

  4. Behavioral Neurosciences Program, Rutgers University, Newark, NJ, USA

    Horacio G. Rotstein

  5. Corresponding Investigators Group, CONICET, Buenos Aires, Argentina

    Horacio G. Rotstein

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  1. Ulises Chialva
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HGR conceived the research and the manuscript. All authors wrote the main manuscript, prepared figures and reviewed the manuscript.

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Correspondence to Horacio G. Rotstein.

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Chialva, U., González Boscá, V. & Rotstein, H.G. Low-dimensional models of single neurons: a review. Biol Cybern 117, 163–183 (2023). https://doi.org/10.1007/s00422-023-00960-1

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