Janet Best

A mathematical model of the sleep/wake cycle.

We present a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle. These features include the timing of sleep and wakefulness under normal and sleep-deprived conditions, ultradian rhythms, more frequent switching between sleep and wakefulness due to the loss of orexin and the circadian dependence of several sleep measures. The model demonstrates how these features depend on interactions between a circadian pacemaker and a sleep homeostat and provides a biological basis for the two-process model for sleep regulation. The model is based on previous “flip–flop” conceptual models for sleep/wake and REM/NREM and we explore whether the neuronal components in these flip–flop models, with the inclusion of a sleep-homeostatic process and the circadian pacemaker, are sufficient to account for the features of the sleep/wake cycle listed above. The model is minimal in the sense that, besides the sleep homeostat and constant cortical drives, the model includes only those nuclei described in the flip–flop models. Each of the cell groups is modeled by at most two differential equations for the evolution of the total population activity, and the synaptic connections are consistent with those described in the flip–flop models. A detailed analysis of the model leads to an understanding of the mathematical mechanisms, as well as insights into the biological mechanisms, underlying sleep/wake dynamics.

Serotonin synthesis, release and reuptake in terminals: a mathematical model

BackgroundSerotonin is a neurotransmitter that has been linked to a wide variety of behaviors including feeding and body-weight regulation, social hierarchies, aggression and suicidality, obsessive compulsive disorder, alcoholism, anxiety, and affective disorders. Full understanding of serotonergic systems in the central nervous system involves genomics, neurochemistry, electrophysiology, and behavior. Though associations have been found between functions at these different levels, in most cases the causal mechanisms are unknown. The scientific issues are daunting but important for human health because of the use of selective serotonin reuptake inhibitors and other pharmacological agents to treat disorders in the serotonergic signaling system.MethodsWe construct a mathematical model of serotonin synthesis, release, and reuptake in a single serotonergic neuron terminal. The model includes the effects of autoreceptors, the transport of tryptophan into the terminal, and the metabolism of serotonin, as well as the dependence of release on the firing rate. The model is based on real physiology determined experimentally and is compared to experimental data.ResultsWe compare the variations in serotonin and dopamine synthesis due to meals and find that dopamine synthesis is insensitive to the availability of tyrosine but serotonin synthesis is sensitive to the availability of tryptophan. We conduct in silico experiments on the clearance of extracellular serotonin, normally and in the presence of fluoxetine, and compare to experimental data. We study the effects of various polymorphisms in the genes for the serotonin transporter and for tryptophan hydroxylase on synthesis, release, and reuptake. We find that, because of the homeostatic feedback mechanisms of the autoreceptors, the polymorphisms have smaller effects than one expects. We compute the expected steady concentrations of serotonin transporter knockout mice and compare to experimental data. Finally, we study how the properties of the the serotonin transporter and the autoreceptors give rise to the time courses of extracellular serotonin in various projection regions after a dose of fluoxetine.ConclusionsSerotonergic systems must respond robustly to important biological signals, while at the same time maintaining homeostasis in the face of normal biological fluctuations in inputs, expression levels, and firing rates. This is accomplished through the cooperative effect of many different homeostatic mechanisms including special properties of the serotonin transporters and the serotonin autoreceptors. Many difficult questions remain in order to fully understand how serotonin biochemistry affects serotonin electrophysiology and vice versa, and how both are changed in the presence of selective serotonin reuptake inhibitors. Mathematical models are useful tools for investigating some of these questions.

Homeostatic mechanisms in dopamine synthesis and release: a mathematical model

BackgroundDopamine is a catecholamine that is used as a neurotransmitter both in the periphery and in the central nervous system. Dysfunction in various dopaminergic systems is known to be associated with various disorders, including schizophrenia, Parkinsons disease, and Tourettes syndrome. Furthermore, microdialysis studies have shown that addictive drugs increase extracellular dopamine and brain imaging has shown a correlation between euphoria and psycho-stimulant-induced increases in extracellular dopamine [1]. These consequences of dopamine dysfunction indicate the importance of maintaining dopamine functionality through homeostatic mechanisms that have been attributed to the delicate balance between synthesis, storage, release, metabolism, and reuptake.MethodsWe construct a mathematical model of dopamine synthesis, release, and reuptake and use it to study homeostasis in single dopaminergic neuron terminals. We investigate the substrate inhibition of tyrosine hydroxylase by tyrosine, the consequences of the rapid uptake of extracellular dopamine by the dopamine transporters, and the effects of the autoreceoptors on dopaminergic function. The main focus is to understand the regulation and control of synthesis and release and to explicate and interpret experimental findings.ResultsWe show that the substrate inhibition of tyrosine hydroxylase by tyrosine stabilizes cytosolic and vesicular dopamine against changes in tyrosine availability due to meals. We find that the autoreceptors dampen the fluctuations in extracellular dopamine caused by changes in tyrosine hydroxylase expression and changes in the rate of firing. We show that short bursts of action potentials create significant dopamine signals against the background of tonic firing. We explain the observed time courses of extracellular dopamine responses to stimulation in wild type mice and mice that have genetically altered dopamine transporter densities and the observed half-lives of extracellular dopamine under various treatment protocols.ConclusionDopaminergic systems must respond robustly to important biological signals such as bursts, while at the same time maintaining homeostasis in the face of normal biological fluctuations in inputs, expression levels, and firing rates. This is accomplished through the cooperative effect of many different homeostatic mechanisms including special properties of tyrosine hydroxylase, the dopamine transporters, and the dopamine autoreceptors.

The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network

A network of excitatory neurons within the pre-Botzinger complex (pre-BotC) of the mammalian brain stem has been found experimentally to generate robust, synchronized population bursts of activity. An experimentally calibrated model for pre-BotC cells yields typical square-wave bursting behavior in the absence of coupling, over a certain parameter range, with quiescence or tonic spiking outside of this range. Previous simulations of this model showed that the introduction of synaptic coupling extends the bursting parameter range significantly and induces complex effects on burst characteristics. In this paper, we use geometric dynamical systems techniques, predominantly a fast/slow decomposition and bifurcation analysis approach, to explain these effects in a two-cell model network. Our analysis yields the novel finding that, over a broad range of synaptic coupling strengths, the network can support two qualitatively distinct forms of synchronized bursting, which we call symmetric and asymmetric bursting, as well as both symmetric and asymmetric tonic spiking. By elucidating the dynamical mechanisms underlying the transitions between these states, we also gain insight into how relevant parameters influence burst duration and interburst intervals. We find that, in the two-cell network with synaptic coupling, the stable family of periodic orbits for the fast subsystem features spike asynchrony within otherwise synchronized bursts and terminates in a saddle-node bifurcation, rather than in a homoclinic bifurcation, over a wide parameter range. As a result, square-wave bursting is replaced by what we call top hat bursting (also known as fold/fold cycle bursting), at least for a broad range of parameter values. Further, spike asynchrony is a key ingredient in shaping the dynamic range of bursting, leading to a significant enhancement in the parameter range over which bursting occurs and an abrupt increase in burst duration as an appropriate parameter is varied.

Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks

Changes in firing patterns are an important hallmark of the functional status of neuronal networks. We apply dynamical systems methods to understand transitions between irregular and rhythmic firing in an excitatory-inhibitory neuronal network model. Using the geometric theory of singular perturbations, we systematically reduce the full model to a simpler set of equations, one that can be studied analytically. The analytic tools are used to understand how an excitatory-inhibitory network with a fixed architecture can generate both activity patterns for possibly different values of the intrinsic and synaptic parameters. These results are applied to a recently developed model for the subthalamopallidal network of the basal ganglia. The results suggest that an increase in correlated activity, corresponding to a pathological state, may be due to an increased level of inhibition from the striatum to the inhibitory GPe cells along with an increased ability of the excitatory STN neurons to generate rebound bursts.

Escape from homeostasis

Many physiological systems, from gene networks to biochemistry to whole organism physiology, exhibit homeostatic mechanisms that keep certain variables within a fairly narrow range. Because homeostatic mechanisms buffer traits against environmental and genetic variation they allow the accumulation of cryptic genetic variation. Homeostatic mechanisms are never perfect and can be destabilized by mutations in genes that alter the kinetics of the underlying mechanism. We use mathematical models to study five diverse mechanisms of homeostasis: thermoregulation; maintenance of homocysteine concentration; neural control by a feed forward circuit; the myogenic response in the kidney; and regulation of extracellular dopamine levels in the brain. In all these cases there are homeostatic regions where the trait is relatively insensitive to genetic or environmental variation, flanked by regions where it is sensitive. Moreover, mutations or environmental changes can place an individual closer to the edge of the homeostatic region, thus predisposing that individual to deleterious effects caused by additional mutations or environmental changes. Mutations and environmental variables can also reduce the size of the homeostatic region, thus releasing potentially deleterious cryptic genetic variation. These considerations of mutations, environment, homeostasis, and escape from homeostasis help to explain why the etiology of so many diseases is complex.

Mathematical Insights into the Effects of Levodopa

Parkinson’s disease has been traditionally thought of as a dopaminergic disease in which cells of the substantia nigra pars compacta (SNc) die. However, accumulating evidence implies an important role for the serotonergic system in Parkinson’s disease in general and in physiological responses to levodopa therapy, the first line of treatment. We use a mathematical model to investigate the consequences of levodopa therapy on the serotonergic system and on the pulsatile release of dopamine (DA) from dopaminergic and serotonergic terminals in the striatum. Levodopa competes with tyrosine and tryptophan at the blood-brain barrier and is taken up by serotonin neurons in which it competes for aromatic amino acid decarboxylase. The DA produced competes with serotonin (5HT) for packaging into vesicles. We predict the time courses of LD, cytosolic DA, and vesicular DA in 5HT neurons during an LD dose. We predict the time courses of DA and 5HT release from 5HT cell bodies and 5HT terminals as well as the changes in 5HT firing rate due to lower 5HT release. We compute the time course of DA release in the striatum from both 5HT and DA neurons and show how the time course changes as more and more SNc cells die. This enables us to explain the shortening of the therapeutic time window for the efficacy of levodopa as Parkinson’s disease progresses. Finally, we study the effects 5HT1a and 5HT1b autoreceptor agonists and explain why they have a synergistic effect and why they lengthen the therapeutic time window for LD therapy. Our results are consistent with and help explain results in the experimental literature and provide new predictions that can be tested experimentally.

Bursts and the Efficacy of Selective Serotonin Reuptake Inhibitors

We present a new hypothesis for the efficacy of selective serotonin reuptake inhibitors (SSRIs). We propose that SSRIs bring the response to the phasic firing of raphe nucleus cells back to normal, even though the average extracellular 5HT concentration remains low. We discuss burst firing in the raphe nuclei and use mathematical models to argue that tonic firing and phasic firing may be decoupled and may come from different mechanisms. We use a mathematical model for serotonin synthesis, release, and reuptake in terminals to illustrate the responses in terminal regions to bursts in a normal individual and in an individual with low vesicular serotonin. We then show that acute doses of SSRIs do not bring the response to bursts back to normal, but that chronic doses do return the response to normal. These model results need to be confirmed by new electrophysiological and pharmacological experiments.

Neuronal models for sleep-wake regulation and synaptic reorganization in the sleeping hippocampus.

In this article, we discuss mathematical models that address the control of sleep-wake behavior in the infant and adult rodent and a model that addresses changes in single-cell firing patterns in the hippocampus across wake and rapid eye movement (REM) sleep states. Each of the models describes the dynamics of experimentally identified neuronal components—either the firing activity of wake-and sleep-promoting neuronal populations or the spiking activity of hippocampal pyramidal neurons. Our discussion of each model illustrates how a mathematical model that describes the temporal dynamics of the modeled neuronal components can reveal specifics about proposed neuronal mechanisms that underlie sleep-wake regulation or sleep-specific firing patterns. For example, the dynamics of the models developed for sleep-wake regulation in the infant rodent lend insight into the involved brain-stem neuronal populations and the evolution of the network during maturation. The results of the model for sleep-wake regulation in the adult rodent suggest distinct properties of the involved neuronal populations and their interactions that account for long-lasting and brief waking bouts. The dynamics of the model for sleep-specific hippocampal neural activity proposes neural mechanisms to account for observed activity changes that can invoke synaptic reorganization associated with learning and memory consolidation.

Models of Dopaminergic and Serotonergic Signaling

Mathematical models of dopaminergic and serotonergic synapses have enabled the authors to study quantitative aspects of the synthesis, release and reuptake of dopamine and serotonin, to investigate the effects of autoreceptors, and to explore the influence of the neurochemistry on the firing patterns of cells known to be involved in the behavioral responses to dopaminergic and serotonergic signaling. The models consist of coupled ordinary differential equations. Parameters are determined from biochemical and physiological measurements. Three results from recent IN SILICO experiments with the dopaminergic and serotonergic synapse models are described: (1) influence of substrate inhibition on the stability of dopamine and serotonin synthesis; (2) a predicted connection between serotonin reuptake transporter (SERT) density on terminals and tonic firing rates; (3) an explanation of data from autoreceptor knock-out experiments. Mathematical models are useful for studying the biology of dopaminergic and serotonergic signaling because these systems are complex and involve interactions between neurochemistry and neurobiology.