Computational modelling of schizophrenic symptoms: basic issues.

Emerging "(computational) systems medicine" challenges neuropsychiatry regarding the development of heuristic computational brain models which help to explore symptoms and syndromes of mental disorders. This methodology of exploratory modelling of mental functions and processes and of their pathology requires a clear and operational definition of the target variable (explanandum). In the case of schizophrenia, a complex and heterogeneous disorder, single psychopathological key symptoms such as working memory deficiency, hallucination or delusion need to be defined first. Thereafter, measures of brain structures can be used in a multilevel view as biological correlates of these symptoms. Then, in order to formally "explain" the symptoms, a qualitative model can be constructed. In another step, numerical values have to be integrated into the model and exploratory computer simulations can be performed. Normal and pathological functioning is to be tested in computer experiments allowing the formulation of new hypotheses and questions for empirical research. However, the crucial challenge is to point out the appropriate degree of complexity (or simplicity) of these models, which is required in order to achieve an epistemic value that might lead to new hypothetical explanatory models and could stimulate new empirical and theoretical research. Some outlines of these methodological issues are discussed here, regarding the fact that measurements alone are not sufficient to build models.

Affective disorders as complex dynamic diseases--a perspective from systems biology.

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.

Systems biology and addiction.

The onset of addiction is marked with drug induced positive experiences that keep being repeated. During that time, adaptation occurs and addiction is stabilized. Interruption of those processes induces polysymptomatic withdrawal syndromes. Abstinence is accompanied by risks of relapse. These features of addiction suggest adaptive brain dynamics with common pathways in complex neuronal networks. Addiction research has used animal models, where some of those phenomena could be reproduced, to find correlates of addictive behavior. The major thrust of those approaches has been on the involvement of genes and proteins. Recently, an enormous amount of data has been obtained by high throughput technologies in these fields. Therefore, (Computational) "Systems Biology" had to be implemented as a new approach in molecular biology and biochemistry. Conceptually, Systems Biology can be understood as a field of theoretical biology that tries to identify patterns in complex data sets and that reconstructs the cell and cellular networks as complex dynamic, self-organizing systems. This approach is embedded in systems science as an interdisciplinary effort to understand complex dynamical systems and belongs to the field of theoretical neuroscience (Computational Neuroscience). Systems biology, in a similar way as computational neuroscience is based on applied mathematics, computer-based computation and experimental simulation. In terms of addiction research, building up "computational molecular systems biology of the (addicted) neuron" could provide a better molecular biological understanding of addiction on the cellular and network level. Some key issues are addressed in this article.

Mathematical model for NF-kappaB-driven proliferation of adult neural stem cells.

Neural stem cells (NSCs) are early precursors of neuronal and glial cells. NSCs are capable of generating identical progeny through virtually unlimited numbers of cell divisions (cell proliferation), producing daughter cells committed to differentiation. Nuclear factor kappa B (NF-kappaB) is an inducible, ubiquitous transcription factor also expressed in neurones, glia and neural stem cells. Recently, several pieces of evidence have been provided for a central role of NF-kappaB in NSC proliferation control. Here, we propose a novel mathematical model for NF-kappaB-driven proliferation of NSCs. We have been able to reconstruct the molecular pathway of activation and inactivation of NF-kappaB and its influence on cell proliferation by a system of nonlinear ordinary differential equations. Then we use a combination of analytical and numerical techniques to study the model dynamics. The results obtained are illustrated by computer simulations and are, in general, in accordance with biological findings reported by several independent laboratories. The model is able to both explain and predict experimental data. Understanding of proliferation mechanisms in NSCs may provide a novel outlook in both potential use in therapeutic approaches, and basic research as well.

Schizophrenia as a dynamical disease.

It is demonstrated that schizophrenia is a dynamical disease, i. e. that important aspects of schizophrenia can be understood on the basis of concepts of the theory of nonlinear dynamical systems. In particular, the gradual shift of a single parameter may result in completely different kinds of behavior of the entire system just like water can be in three qualitatively different states depending on the single parameter temperature. Transitions between these different states are called bifurcations or phase transitions. In case of schizophrenia the parameter may be the neurotransmitter dopamine (or serotonin or glutamate). With a high level of dopamine transmission the symptoms of schizophrenia appear, and with rather low levels those of Morbus Parkinson. In between a healthy state prevails. With respect to neuronal activity the effects of dopamine are demonstrated by a mathematical model which can be interpreted in two ways: on the one hand as a model of typical excitatory-inhibitory circuits in the cortex, on the other hand of a negative feedback loop between thalamus, prefrontal cortex and striatum. The model exhibits different types of firing patterns and their bifurcation, from various kinds of periodicities up to erratic or chaotic behavior, corresponding to different levels of dopamine concentration.

A theoretical approach to G-protein modulation of cellular responsiveness.

Structure and function of cells often depend critically on molecular signals arriving at their surface. There are universal mechanisms of signal transduction and signal processing across cell membranes. In this paper the mechanisms involving guanine-nucleotide regulatory proteins ("G-proteins") and certain receptor-kinases are considered. On the basis of recent findings in molecular biology a mathematical model is developed taking into account all essential components in the biochemical network between first and second messenger. There are two coupled feedback loops inherent in this process. The model finally consists of three nonlinear equations, which are obtained from a system of originally ten equations by using conservation laws and quasi-steady state conditions. The second part of the paper contains a mathematical analysis of the model. Solutions describing the temporal development of the involved biochemical species are shown to be bounded, more specifically to remain, independent of the size of the input signal, in a bounded domain of the state space. For the situation of stationary input signals existence, uniqueness and asymptotic stability of steady states are derived. We also demonstrate biologically relevant stimulus-response properties like monotonicity and saturation effects. For temporally non-constant input signals we show numerically that the model is able to produce phenomena of hypersensitivity and desensitization which are important characteristics of cellular responsiveness.

Circadian aspects of apparent correlation dimension in human heart rate dynamics.

The purpose of this study was to examine changes in complexity of cardiac dynamics over 24 h. With use of Holter monitoring, 27 24-h electrocardiogram recordings were obtained from 15 healthy subjects. For each recording, the apparent dimension (DA) was calculated for consecutive sections of 500 heartbeats. These were used to determine nighttime and daytime dimension (D(An) and D(Ad), respectively) as well as the difference between D(An) and D(Ad) (delta DA). Mean 24-h DA, D(An), and D(Ad) were 5.9 +/- 0.3, 6.3 +/- 0.5, and 5.6 +/- 0.6, respectively. D(An) was significantly higher than D(Ad) (P < 0.001), with a mean delta DA of 0.6 +/- 0.7. Furthermore, 67% of delta DA values were significantly different from zero at the 0.05 level. The results show that dimension analysis may be applied to heart rate dynamics to reveal circadian differences of heart rate complexity. We suggest that the decreased complexity during daytime may result from the synchronization of physiological functions. The increase in complexity at night would then correspond to an uncoupling of these functions during the regenerative period.

A basic mathematical model of the immune response.

Interaction of the immune system with a target population of, e.g., bacteria, viruses, antigens, or tumor cells must be considered as a dynamic process. We describe this process by a system of two ordinary differential equations. Although the model is strongly idealized it demonstrates how the combination of a few proposed nonlinear interaction rules between the immune system and its targets are able to generate a considerable variety of different kinds of immune responses, many of which are observed both experimentally and clinically. In particular, solutions of the model equations correspond to states described by immunologists as "virgin state," "immune state" and "state of tolerance." The model successfully replicates the so-called primary and secondary response. Moreover, it predicts the existence of a threshold level for the amount of pathogen germs or of transplanted tumor cells below which the host is able to eliminate the infectious organism or to reject the tumor graft. We also find a long time coexistence of targets and immune competent cells including damped and undamped oscillations of both. Plausibly the model explains that if the number of transformed cells or pathogens exeeds definable values (poor antigenicity, high reproduction rate) the immune system fails to keep the disease under control. On the other hand, the model predicts apparently paradoxical situations including an increased chance of target survival despite enhanced immune activity or therapeutically achieved target reduction. A further obviously paradoxical behavior consists of a positive effect for the patient up to a complete cure by adding an additional target challenge where the benefit of the additional targets depends strongly on the time point and on their amount. Under periodically pulsed stimulation the model may show a chaotic time behavior of both target growth and immune response. (c) 1995 American Institute of Physics.

Mathematical model and simulation of retina and tectum opticum of lower vertebrates.

The processing of information within the retino-tectal visual system of amphibians is decomposed into five major operational stages, three of them taking place in the retina and two in the optic tectum. The stages in the retina involve (i) a spatially local high-pass filtering in connection to the perception of moving objects, (ii) separation of the receptor activity into ON- and OFF-channels regarding the distinction of objects on both light and dark backgrounds, (iii) spatial integration via near excitation and far-reaching inhibition. Variation of the spatial range of excitation and inhibition allows to account for typical activities observed in a variety of classes of retina ganglion cells. Mathematical description of the operations in the tectum opticum include (i) spatial summation of retinal output (mainly of class-2 and class-3 retina ganglion cells), and (ii) direct or indirect lateral inhibition between tectal cells. In the computer simulation, first the output of the mathematical retina model is computed which, then, is used as the input to the tectum model. The full spatio-temporal dynamics is taken into account. The simulations show that different combinations of strength of lateral inhibition on the one side and the response properties of the retina ganglion cells on the other side determine the response properties of tectal cell types involved in object recognition.

Principles of self-generation and self-maintenance.

Living systems are characterized as self-generating and self-maintaining systems. This type of characterization allows integration of a wide variety of detailed knowledge in biology. The paper clarifies general notions such as processes, systems, and interactions. Basic properties of self-generating systems, i.e. systems which produce their own parts and hence themselves, are discussed and exemplified. This makes possible a clear distinction between living beings and ordinary machines. Stronger conditions are summarized under the concept of self-maintenance as an almost unique character of living systems. Finally, we discuss the far-reaching consequences that the principles of self-generation and self-maintenance have for the organization, structure, function, and evolution of single- and multi-cellular organisms.

The dynamics of recurrent inhibition.

A heuristic model for the dynamics of recurrent inhibition, emphasizing non-linearities arising from the stoichiometry of transmitter-receptor interactions and time delays due to finite feedback pathway transmission times, is developed and analyzed. It is demonstrated that variation in model parameters may lead to the existence of multiple steady states, and the local stability of these are analyzed as well as the occurrence of switching behaviour between them. As an example of the applicability of this model, parameters are estimated for the hippocampal mossy fibre-CA3 pyramidal cell-basket cell complex. Numerically simulated responses of this system to alterations in presynaptic drive and titration of inhibitory transmitter receptors by penicillin are presented. Numerical simulations indicate the existence of multiple bifurcations between periodic solutions, as well as the existence of bifurcations to chaotic solutions, as presynaptic drive and receptor density are varied. It is hypothesized that the model offers insight into the sequences of events recorded in single CA3 pyramidal cells following the application of penicillin, a specific inhibitory receptor blocking agent.

Delays in physiological systems.

In comparison to most physical or chemical systems, biological systems are of extreme complexity. In addition the time needed for transport or processing of chemical components or signals may be of considerable length. Thus temporal delays have to be incorporated into models leading to differential-difference and functional differential equations rather than ordinary differential equations. A number of examples, on different levels of biological organization, demonstrate that delays can have an influence on the qualitative behavior of biological systems: The existence or non-existence of instabilities and periodic or even chaotic oscillations can entirely depend on the presence of absence of delays with appropriate duration.