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A mathematical model that contains two pulse generators based on integrate-and-fire mechanism - one located in the hypothalamus, the other in the pituitary gland. The target is a periphery gland.

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Ultradian_pulse_generators

This MATLAB program was designed for generating time series and figures for the talk

Alexander N. Churilov and John G. Milton "An Integrate-and-fire Mechanism for Modeling Rhythmicity in the Neuroendocrine System"

presented at the international conference Big Brain 2022, Fudan University, Shanghai, China, November/December 2022. Conference proceedings to be published by De Gruyter.

Setting

We study a new mathematical model of a hormonal axis that comprises two coupled ultradian pulse generators. One generator is located in the hypothalamus, the second is located in the anterior pituitary. Two integrate-and-fire schemes are used to describe the pulse generator mechanisms. Depending on their firing thresholds and on the coupling gains, the system exhibits a variety of periodic or quasi-periodic behaviors.

Schematic representation of a hormonal axis

Arrows and bar-headed lines indicate excitatory and inhibitory connections, respectively. Here $H(t)$ is the input from the suprachiasmatic nucleus (SCN) of the hypothalamus, $x(t)$, $y(t)$, $z(t)$ are serum concentrations of the hypothalamic, pituitary and target gland hormones, respectively.

Integrate-and fire model of a single peptide hormone's release

Let $x(t)$ be the serum concentration of a peptide hormone,

$$\dot x = -\alpha x(t) +S(t)$$

with the clearance coefficient $\alpha>0$ and the secretion rate given by a function $S(t)$.

Let $V(t)$ be an impulsive membrane potential. The pulsation times $t_n$ are defined from

$$t_0=0, \quad t_{n+1} = \min\{t \;:\; t>t_n,\quad V(t)=\Delta\}.$$

where $\Delta>0$ a given threshold. After the impulse, the potential resets to zero, i.e.

$$V(t_n^+) = 0,\quad n\ge 0,$$

Between the impulses in membrane potential satisfies the differential equation

$$\dot V = -\mu (V(t) - V_0) + I(t),$$

where $I(t)$ is a consolidated input of the considered hormonal gland from some other organs, $\mu$ and $V_0$ are positive coefficients.

The secretion rate $S(t)$ is a functional of $I(t)$ and $V(t)$, namely

$$S(t)= k (I(t) + I_0) F(V(t)),$$

where $k$, $I_0$ are positive constants and the $F(V)$ is a shaping function,

$$F(V) = \lambda V\,\exp(-\lambda V +1).$$

Here $\lambda>0$ is a constant parameter.

These equations define a mapping

$$G_{p}\;:\; I(t) \mapsto x(t)$$

with a vector of parameters

$$p = \{\, I_0,\; V_0,\; \alpha,\; \lambda,\; \mu,\; k, \; \Delta \,\}.$$

Mathematical model of a regulation loop consisting of three hormones

Let $x(t)$, $y(t)$, $z(t)$ be serum concentrations of the hypothalamic, pituitary and target gland hormones, respectively.

Hypothalamic pulse generator.

Use a pulse generator described in the previous section with a vector of parameters $p_1$:

$$G_{p_1}\;:\; I_1(t) \mapsto x(t),\quad p_1 = \{\, I_0^{(1)},\; V_0^{(1)},\; \alpha_1,\; \lambda_1,\; \mu_1,\; k_1, \; \Delta_1 \,\}.$$

The input function $I_1(t)$ is

$$I_1(t) = (1+H(t))\, L_1(z(t)).$$

\end{equation} and contains two components: a modulating input, $H(t)$, and an inhibitory input, $L_1(z(t))$. The feedback function $L_1(z)$ obeys Michaelis-Menten kinetics

$$L_1(z) = \frac{1}{1 + z/h_1},$$

where $h_1>0$ is a parameter. The function $H(t)$ is the modulating input from the suprachiasmatic nucleus of the hypothalamus. In the simplest case it can be chosen harmonic.

Pituitary pulse generator.

Consider the pulse generator with a vector of parameters $p_2$

$$G_{p_2}\;:\; I_2(t) \mapsto y(t),\quad p_2 = \{\, I_0^{(2)},\; V_0^{(2)},\; \alpha_2,\;\lambda_2,\; \mu_2,\; k_2, \; \Delta_2 \,\}.$$

The input function $I_2(t)$ is

$$I_2(t) = x(t)\, L_2(z(t)),$$

where $x(t)$ is an excitatory input and and $L_2(z(t))$ is an inhibitory input described by a decreasing positive function, which can also be taken Michaeles-Menten.

Target gland hormonal release.

Suppose that the target hormone is released continuously, following a linear differential equation

$$\dot z = -\alpha_3 z + k_3 y,$$

where $\alpha_3$, $k_3$ are positive parameters.

Simulations

The depository contains a MATLAB program for simulating and drawing hormonal profiles. The simulations are illustrated with figures pic01.png, pic02.png, pic03.png given in Images folder.


Hormonal profiles for isolated hypothalamic and pituitary hormonal generators. Hormonal profiles for isolated hypothalamic and pituitary hormonal generators.


Hormonal profiles for the lesioned circadian input. Hormonal profiles for the lesioned circadian input.


Hormonal profiles for the circadian input. Hormonal profiles for the circadian input.


Our previous publications on integrate-and-fire models

  1. A. N. Churilov, J. Milton, and E. R. Salakhova. An integrate-and-fire model for pulsatility in the neuroendocrine system. Chaos (AIP journal), 30(8):083132, 2020.

  2. A. N. Churilov and J. G. Milton. Modeling pulsativity in the hypothalamic-pituitary-adrenal hormonal axis. Scentific Reports, 12:8480, 2022.

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A mathematical model that contains two pulse generators based on integrate-and-fire mechanism - one located in the hypothalamus, the other in the pituitary gland. The target is a periphery gland.

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