Mathematical modeling of zebrafish social behavior in response to acute caffeine administration

Abstract

Zebrafish is a model organism that is receiving considerable attention in preclinical research. Particularly important is the use of zebrafish in behavioral pharmacology, where a number of high-throughput experimental paradigms have been proposed to quantify the effect of psychoactive substances consequences on individual and social behavior. In an effort to assist experimental research and improve animal welfare, we propose a mathematical model for the social behavior of groups of zebrafish swimming in a shallow water tank in response to the administration of psychoactive compounds to select individuals. We specialize the mathematical model to caffeine, a popular anxiogenic compound. Each fish is assigned to a Markov chain that describes transitions between freezing and swimming. When swimming, zebrafish locomotion is modeled as a pair of coupled stochastic differential equations, describing the time evolution of the turn-rate and speed in response to caffeine administration. Comparison with experimental results demonstrates the accuracy of the model and its potential use in the design of in-silico experiments.

Keywords: Anxiety; Danio rerio; collective behavior; in-silico; pharmacology; social interaction; stochastic differential equations.

Figure 1.

Representative trajectories of a group of five subjects, with four untreated individuals and one subject treated with caffeine, at a concentration of: 0 (A), 25 (B), 50 (C), and 70 mg/L (D).

Figure 2.

(A) Fish kinematics: at time t, the fish pose is denoted as [x_i(t), y_i(t), θ_i(t)]^T, swimming at speed ν_i(t) and turn-rate ω_i(t) (B) A close-up look at the interaction between a pair of fish within a group of hve fish. Alignment and attraction between the i- and j-th fish are functions of the relative orientation, ϕ_ij(t), and relative position, in terms of the distance between fish, S_ij(t), and relative angle, θ_ij(t).

Figure 3.

Comparisons of discrete-time Markov chain parameters of the treated fish across caffeine concentrations and social environment (single or group). The bars represent the mean value of the probability of transition from freezing to swimming (A), and the mean value of the probability of transition from swimming to freezing (B). The striped bars correspond to the calibrated parameters for the case of a single treated fish from Burbano-Lombana and Porfiri (2020). The solid bars correspond to the calibrated parameters for the case of a treated fish swimming in a social group. The vertical red error bars represent standard errors of the means.

Figure 4.

Comparisons of the locomotion parameters corresponding to the speed evolution of the treated fish across caffeine concentrations and social environment (single or group). The bars represent the mean value of the linear expansion rate (A), and strength of added noise in the speed evolution (B). The striped bars correspond to the calibrated parameters for the case of a single treated fish from Burbano-Lombana and Porfiri (2020). The solid bars correspond to the calibrated parameters for the case of a treated fish swimming in a social group. The symbol $ indicates a significant difference (p < 0.050) in Tukey’s HSD post-hoc analysis comparing individuals swimming alone or on group (single versus group). The vertical red error bars represent standard errors of the means.

Figure 5.

Comparisons of the locomotion parameters corresponding to the turn-rate evolution of the treated fish across caffeine concentrations, and social environment (single or group). The bars represent the mean value of the mean reversion rate (A), strength of added noise in the turn-rate evolution (B), intensity of jumps in the turn-rate evolution (C), and frequency of jumps in turn-rate evolution (D). The striped bars correspond to the calibrated parameters for the case of a single treated fish from Burbano-Lombana and Porfiri (2020). The solid bars correspond to the calibrated parameters for the case of a treated fish swimming in a social group. Different letters on top of the bars indicate a significant difference (p < 0.050) in Tukey’s HSD post-hoc analysis across caffeine concentrations, comparing individuals swimming in isolation (standard font) or in group (italic font). The symbol $ indicates a significant difference (p < 0.050) in Tukey’s HSD post-hoc analysis comparing individuals swimming alone or on group (single versus group). The vertical red error bars represent standard errors of the means.

Figure 6.

Calibrated interaction parameters in the turn-rate evolution across caffeine concentrations. The bars represent the mean value of the attraction gain of treated fish towards untreated fish (A), alignment gain of treated fish towards untreated fish (B), average attraction gain of untreated fish towards treated fish (C), average alignment gain of untreated fish towards treated fish (D), average attraction gain between untreated fish (E), and average alignment gain between untreated fish (F). Different letters on top of the bars indicate a significant difference (p < 0.050) in Tukey’s HSD post-hoc analysis across caffeine concentrations. The vertical red error bars represent standard errors of the means.

Figure 7.

Representative in-silico trajectories of a group of five subjects, with four untreated individuals and one subject treated with caffeine, at a concentration of: 0 (A), 25 (B), 50 (C), and 70 mg/L (D).

Figure 8.

Comparisons of the schooling tendency of the fish, measured in terms of average polarization (A), and average relative polarization (B), across caffeine concentrations and data-types (experiment or in-silico). Different letters on top of the bars indicate a significant difference (p < 0.050) in Tukey’s HSD post-hoc analysis across caffeine concentrations, comparing interaction metrics in experiment (standard font) or in-silico (italic font) data-type. The vertical red error bars represent standard errors of the means.

Figure 9.

Comparisons of the shoaling tendency of the fish, measured in terms of the average distance between treated fish and untreated fish (A), and average distance between untreated fish (B), across caffeine concentrations and data-types (experiment or in-silico). Different letters on top of the bars indicate a significant difference (p < 0.050) in Tukey’s HSD post-hoc analysis across caffeine concentrations, comparing interaction metrics in experiment (standard font) or in-silico (italic font) data-type. The vertical red error bars represent standard errors of the means.