Affective Disorders as Complex Dynamic Diseases – a Perspective from Systems Biology

 

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Abstract

Understanding mental disorders and their neurobiological basis encompasses the conceptual management of “complexity” and “dynamics”. For example, affective disorders exhibit several fluctuating state variables on psychological and biological levels and data collected of these systems levels suggest quasi-chaotic periodicity leading to use concepts and tools of the mathematics of nonlinear dynamic systems. Regarding this, we demonstrate that the concept of “Dynamic Diseases” could be a fruitful way for theory and empirical research in neuropsychiatry. In a first step, as an example, we focus on the analysis of dynamic cortisol regulation that is important for understanding depressive disorders. In this case, our message is that extremely complex phenomena of a disease may be explained as resulting from perplexingly simple nonlinear interactions of a very small number of variables. Additionally, we propose that and how widely used complex circuit diagrams representing the macroanatomic structures and connectivities of the brain involved in major depression or other mental disorders may be “animated” by quantification, even by using expert-based estimations (dummy variables). This method of modeling allows to develop exploratory computer-based numerical models that encompass the option to explore the system by computer simulations (in-silico experiments). Also inter- and intracellular molecular networks involved in affective disorders could be modeled by this procedure. We want to stimulate future research in this theoretical context.

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Appendix

The model contains two time dependent variables:

x (t)=concentration of ACTH in the blood at time t,

y (t)=concentration of cortisol in the blood at time t.

The rates of change of these two state variables are denoted by

respectively (mathematically speaking these are the derivatives of x and y with respect to t, or the velocities by which these concentrations change).

The fact that the increase of cortical concentration is the greater the greater the ACTH-concentration, is expressed by the equation

where g (x) is a function (in the mathematical sense) such that the larger x, the larger is g (x). An example of such a function is g (x)=dx ^m /(k+x ^m ) with positive constant (with respect to time) parameters d, m, k.

Another constant parameter is τ which denotes the length of the time interval between the release of ACTH (=x) from the pituitary until the release of cortisol (y) from the adrenal gland into the blood. The cortisol molecules decay with a rate constant a, and hence the above equation for the change of y is to be extended to

Not very differently we can propose an equation for the rate of change of ACTH:

However, since cortisol (y) inhibits (both across the hypothalamus and the pituitary) the production of ACTH (x), the function f (y) must be decreasing, i. e., the higher the value of y the lower the value of f (y). An example of such a function is given by

with positive constant parameters c, K, n.

The mathematical model consisting of the two equations (1) and (2) is called in mathematical terms a “nonlinear system of two differential delay equations”. It is nonlinear because the gain functions f and g are nonlinear. It contains the two delays δ and τ, however, without loss of generality δ=0 can be assumed.

The interactive structure described by the equations (1) and (2) allows essentially for two kinds of dynamics: (i) a stable equilibrium, i. e., temporally constant solutions x (t)=constant, y (t)=constant such that after any perturbation of x or y the system returns to this equilibrium; (ii) a stable periodic oscillator (sometimes called limit cycle), i. e., both x (t) and y (t) increase and decrease in a periodic fashion and, moreover, after external perturbations away from this periodicity the system returns to the same periodic behavior (to a stable clock). Whether the situation (i) or alternatively (ii) occurs depends only on the values of the temporally constant parameters of the model.

More complex behavior arises when this oscillatory system is coupled to a second oscillator (“coupled oscillators”). This is definitely the case since it is well known that the activity of the hypothalamus is influenced by the suprachiasmatic nucleus, a tiny region of the brain considered to be the origin of the circadian rhythm. This rhythmic influence is taken into account in the mathematical model by including a periodic function into the differential equation (2):

Now the model consists of equation (3) together with equation (1). From the point of view of systems theory the operational structure of this system is that of 2 coupled oscillators. In [Fig. 2d] a solution of this system is presented. At least qualitatively there is coincidence with the rather irregular (quasi-chaotic) empirical data shown in [Fig. 2c]. Another interpretation of the term A sin (ω t) could be that the pituitary is in itself an autonomous oscillator mimicked by this term. The pattern observable in [Fig. 2d] is just one of a large variety of qualitatively different solution types of System (2)−(3) including different ratios of phase locking and different types of chaos (for more examples see an der Heiden 1992 [29]). Including the effect of the immune system sketched in the diagram of [Fig. 2a] and/or other influences from the brain into the mathematical model presented here would still substantially increase the complexity of its behavior.

Correspondence

Prof. Dr. Dr. Dr. F. Tretter

Department of Addiction

Isar Amper Clinics

Ringstraße 9

85540 Haar/Munich

Germany

Email: felix.tretter@iak-kmo.de