Home

ultradian_pulse_generators

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

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

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.