Description and Validation of a Dynamical Systems Model of Presynaptic Serotonin Function: Genetic Variation, Brain Activation and Impulsivity

Abstract

Despite more than a decade of empirical work on the role of genetic polymorphisms in the serotonin system on behavior, the details across levels of analysis are not well understood. We describe a mathematical model of the genetic control of presynaptic serotonergic function that is based on control theory, implemented using systems of differential equations, and focused on better characterizing pathways from genes to behavior. We present the results of model validation tests that include the comparison of simulation outcomes with empirical data on genetic effects on brain response to affective stimuli and on impulsivity. Patterns of simulated neural firing were consistent with recent findings of additive effects of serotonin transporter and tryptophan hydroxylase-2 polymorphisms on brain activation. In addition, simulated levels of cerebral spinal fluid 5-hydroxyindoleacetic acid (CSF 5-HIAA) were negatively correlated with Barratt Impulsiveness Scale (Version 11) Total scores in college students (r = −.22, p = .002, N = 187), which is consistent with the well-established negative correlation between CSF 5-HIAA and impulsivity. The results of the validation tests suggest that the model captures important aspects of the genetic control of presynaptic serotonergic function and behavior via brain activation. The proposed model can be: (1) extended to include other system components, neurotransmitter systems, behaviors and environmental influences; (2) used to generate testable hypotheses.

Appendix

Berkeley Madonna Code to simulate differential and difference Eqs. 8–17:

$$ \begin{aligned}& {\text{METHOD}}\;{\text{ RK}}4 \\& {\text{STARTTIME}} = 0 \\& {\text{STOPTIME}} = 200 \\& {\text{DT}} = 0.01 \\& {\text{d}}/{\text{dt}}({\text{x}}1)=-({\text{g}}+{\text{d}})*{\text{x}}1+{\text{u}}*(1-\exp(-2*{\text{x}}2))/(1+\exp(-2*{\text{x}}2)) \\& {\text{INIT}}\;{\text{x}}1=0 \\& {\text{d}}/{\text{dt}}({\text{x}}2)={\text{g}}*{\text{x}}1-{\text{k}}4*{\text{x}}2+{\text{r}}-{\text{u}} \\& {\text{INIT}}\;{\text{x}}2=1 \\& {\text{d}}/{\text{dt}}({\text{c}})=-{\text{c}}+(({\text{k}}\_1/{\text{k}}1{\text{trp}})+{\text{c}})*{\text{y}}1+{\text{v}} \\& {\text{INIT}}\;{\text{c}}=1 \\& {\text{d}}/{\text{dt}}({\text{y}}1)=({\text{c}}-(({\text{k}}\_1 + {\text{k}}2{\text{trp}})/{\text{k}}1{\text{trp}}+{\text{c}})*{\text{y}}1)/{\text{e}} \\& {\text{INIT}}\;{\text{y}}1=1 \\& {\text{d}}/{\text{dt}}({\text{r}})=-{\text{k}}3*{\text{r}}+{\text{k}}2{\text{trp}}*{\text{y}}1 \\& {\text{INIT}}\;{\text{r}} = 1 \\& {\text{d}}/{\text{dt}}({\text{l}})=-{\text{l}}+(({\text{k}}\_1/{\text{k}}1{\text{maoa}})+{\text{l}})*{\text{z}}1+{\text{k}}4*{\text{x}}2 \\& {\text{INIT}}\;{\text{l}} = 1 \\& {\text{d}}/{\text{dt}}({\text{z}}1)=({\text{l}}-(({\text{k}}\_1+{\text{k}}2{\text{maoa}})/{\text{k}}1{\text{maoa}}+{\text{l}})*{\text{z}}1)/{\text{e}} \\& {\text{INIT}}\;{\text{z}}1 = 0 \\& {\text{d}}/{\text{dt}}({\text{p}})=-{\text{k}}3*{\text{p}}+{\text{k}}2{\text{maoa}}*{\text{z}}1 \\& {\text{INIT}}\;{\text{p}} = 0 \\& {\text{rand}}={\text{RANDOM}}(0,1) \\& {\text{u}}={\text{IF}}\;({\text{SIN}}({\text{TIME}}/{\text{PER}})>0)\;{\text{THEN}}\;({\text{IF}}\;({\text{rand}}>1-{\text{aprob}})\;{\text{AND}}\;(({\text{x}}1-{\text{x}}1{\text{th}})>0)\;{\text{THEN}}\;0\;{\text{ELSE}}\;({\text{Rmax}}-{\text{b}}*{\text{x}}1)*{\text{x}}2)\;{\text{ELSE}}\;0 \\& {\text{INIT}}\;{\text{Fire Rate}}=0 \\& {\text{INIT}}\,{\text{Fire}} = 0 \\& {\text{FireRate}}({\text{t}}+{\text{Dt}})={\text{FireRate}}+{\text{Fire}} \\& {\text{Next}}\;{\text{Fire}}={\text{IF}}\;{\text{u}}>0\;{\text{THEN}}\;1\;{\text{ELSE}}\;0 \\\end{aligned} $$

See Table 1.

Table 1 Model parameter values