Neural and Neuroendocrine Mathematical Models

Mathematical models and mathematical analysis from our lab and other labs for neurons and neuron-like endocrine cells in the anterior pituitary are listed below.

View other models from our lab by subject on our Mathematical Models page. Visit to view a list of models by publication citation.

Adaptation and Low-Frequency Firing

Low-frequency firing in neurons occurs near SNIC bifurcations, but we show that this feature is neither necessary nor sufficient using models based on Hindmarsh-Rose and Morris-Lecar. More fundamental is an adaptation current that stretches out the frequency-current curve and makes low-frequency activity robust.

How Adaptation Makes Low Firing Rates Robust.
Ha J, Sherman A.
J. Math. Neurosci. (2017) 7:4. Abstract/Full Text
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Generic Pituitary Model

The anterior pituitary contains six endocrine cells each secreting a unique hormone. We show with a generic model that a core set of ionic currents unites the whole family, while other currents provide the distinguishing features that enable them to do their specific jobs.

Common and diverse elements of ion channels and receptors underlying electrical activity in endocrine pituitary cells.
Fletcher PA, Sherman A, Stojilkovic SS.
Molecular and Cellular Endocrinology. (2018) 463:23-36 Abstract/Full Text
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Morris-Lecar Model

This is a simplified Hodgkin-Huxley-type model. Originally meant to model barnacle muscle, it has become a standard tool for mathematical analysis of neural models. The fast electrical core of our beta-cell models is based on Morris-Lecar.

Analysis of Neural Excitability and Dynamics.
Ermentrout B, Rinzel J.
“Methods in Neuronal Modeling.” Abstract/Full Text
Voltage oscillations in the barnacle giant muscle fiber.
Lecar H, Morris C.
Biophys. J. (1981) 35(1):193-213. Abstract/Full Text
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Phase Resetting in a Model for Pituitary Bursting

Two different classes of model for square-wave bursting in pituitary cells can be distinguished by their responses to resetting from the silent to the active phase. The model behavior resembles that seen in somatotrophs, lactotrophs, and corticotrophs.

Resetting behavior in a model of bursting in secretory pituitary cells: Distinguishing plateaus from pseudo-plateaus.
LeBeau A, Osinga HM, Sherman A, Stern JV.
Bull. Math. Biol. (2007) 70(1):68-88 Abstract/Full Text
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Pituitary Corticotroph Model

This model demonstrates that the diverse electrical patterns, including spiking and bursting, seen in pituitary corticotrophs in response to corticotropin releasing hormone and arginine vasopressin can be accounted for by variation in as few as two parameters.

Modeling the diversity of spontaneous and agonist-induced electrical activity in anterior pituitary corticotrophs.
Fletcher PA, Sherman A, Stojilkovic SS, Zemkova H.
J. Neurophysiol. (2017) 117(6):2298-2311 Abstract/Full Text
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Spontaneous and Receptor-Controlled Electrical Activity in Pituitary Somatotrophs

This model accounts for the natural variation and responses to ion substitution and modulation by GHRH and somatostatin seen in pituitary somatotrophs.

Mechanism of spontaneous and receptor-controlled electrical activity in pituitary somatotrophs: Experiments and theory.
Sherman A, Stojilkovic SS, Tsaneva-Atanasova K, van Goor F.
J. Neurophysiol. (2007) 98(1):131-44 Abstract/Full Text
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