Analysis of Group of Fish Response to Startle Reaction

Abstract

In animal groups, startle reaction is classified among alarm reactions observed during anti-predatory or chemical reactions in the context of fear-inducing stimulation. For fish groups, this behavior often initiated by a few individuals in response to a strong stimulus is characterized by sudden and abrupt turns coupled to increasing speed that can initially disturb the group, but eventually increase group coordination as time evolves. Departing from these observations, we leverage a model of fish swimming to recreate startle reaction in terms of a biased stochastic jump diffusion process and to study their effects on the collective response of groups of fish. We contrast the effectiveness of the modeled startle reaction against a traditional approach to recreate leadership to find that for specific range values of occurrence rate and intensity, the fast turns initiated by a few individual can divert followers from their reference trajectory while keeping them aligned and in a closer proximity with the startled fish. For small noise perturbations, we propose a closed-form expression for the polarization order parameter which provides a good prediction of the group alignment as a function of the noise parameters. These findings offer evidence that startle reaction can be utilized to consistently recreate the emergence of leadership observed in animal groups with potential applications to multi-agent systems.

Appendix

1.1 Alternative Distributions to Recreate Startle Response

See Fig. 10.

Fig. 10

Group response measured by group alignment (first row) and group cohesion (second row) by varying the jump frequency $\iota$ and intensity $\delta$ for two additional distributions (the right shifted normal distribution and the uniform shifted distribution) with identical mean and variance as the half-normal distribution. Group size is $N=10$ fish and a $100 \times 100$ grid size is considered for $\iota$ and $\delta$. Model parameters are in Table 1

1.2 Stability Analysis

We study here group response to small perturbations introduced at a closer proximity of the coordinated state when all particles tend to align their heading angle with a reference state. Similar to Calovi et al. (2015), the coupled dynamical system defined in Eqs. (1) and (3) can be compared to particles interacting through attractive and repulsive forces which tend to align them in a direction prescribed by the informed fish acting as an attraction field. Introducing the polarization P measuring the degree of order in the system, group response to small perturbations denoted P-susceptibility is equivalent up to a constant factor to the fluctuation of the order parameter P (Calovi et al. 2015; Marconi et al. 2008):

$$\begin{aligned} N\left[ \langle P^2\rangle - \langle P\rangle ^2\right] = \frac{\partial P}{\partial x }_{|x=0}, \end{aligned}$$

(13)

where $\langle P\rangle = \lim _{T\rightarrow \infty } 1/(T-T_r) \sum _{k=T_r}^{T} P(k)$, and x is associated with the perturbation field. In our system, for a fixed value of $\varsigma$, the perturbation field is essentially characterized by the jumps parameters $\iota$ and $\delta$. We seek to establish a closed-form expression to quantify group response measured by the alignment order parameter to small variations of $\iota$ and $\delta$.

We note that in a closer proximity of the coordinated state, all individuals share a quasi-identical heading direction such that $\theta _{ij} \simeq 0$ for all i, j. In addition, the order parameter (proposed in (9) in terms of the cosine similarity) is identical to the polarization order parameter ($\mathrm {Pol}$), that is, $\lim _{k\rightarrow \infty } \mathbf {E}\left[ P(k)\right] =\lim _{k\rightarrow \infty } \mathbf {E}\left[ \mathrm {Pol}\right]$ at a closer proximity of the coordinated state. We leverage a mean square stability study of a similar stochastic double integrator introduced in Mwaffo et al. (2015b), Mwaffo and Porfiri (2016) to propose a closed-form expression of the polarization order parameter of the system in (7). To proceed, we introduce $\tilde{w}_i(k) = w_i(k) - w_i^\star (k)$ in order to transform (7) into the following system:

$$\begin{aligned} \begin{aligned} \theta _i(k+1)&= \theta _i(k) + w_i^\star (k)\Delta \tau + \tilde{w}_i(k)\Delta \tau \\ \tilde{w}_i(k+1)&= \tilde{w}_i(k) e^{- \Delta \tau } + \varsigma \sqrt{\frac{1}{2}\left( 1-e^{-2 \Delta \tau }\right) } \varepsilon _i(k) + \delta \Delta \nu _i(k\Delta \tau ) \zeta _i(k), \end{aligned} \end{aligned}$$

(14)

where $w_i^\star (k) = w_i^\star (\tau _k)$ is defined in (4b). Now considering that $\Delta \tau$ is small enough, the above system can be reduced to:

$$\begin{aligned} \begin{aligned} \theta _i(k+1)&= \theta _i(k) + w_i^\star (k)\Delta \tau + \tilde{w}_i(k+1) \Delta \tau \\ \tilde{w}_i(k+1)&= (1-\Delta \tau )\tilde{w}_i(k) + \varsigma \sqrt{\Delta \tau } \varepsilon _i(k) + \delta \Delta \nu _i(k\Delta \tau ) \zeta _i(k), \end{aligned} \end{aligned}$$

(15)

where $w_i^\star (k) = w_i^\star (\tau _k)$ is defined in (4b). Considering small misalignment ($\theta _{ij} \ll 1$) between pair of fish i and j, one can estimate $\sin \left( \theta _{ij}(k)\right) \simeq \theta _{ij}(k)$ and $\phi _{ij}(k)\simeq \frac{\pi }{2}$. This allows us to obtain the following discrete-time double integrator system:

$$\begin{aligned} \begin{aligned} \theta _i(k+1)&= \theta _i(k) + \kappa _v\Delta \tau \sum _{j\in \mathcal {N}_i(k)}^{} \theta _{ij}(k) + \tilde{w}(k)\Delta \tau \\ \tilde{w}_i(k+1)&= \left( 1-\Delta \tau \right) \tilde{w}_i(k) + \varsigma \sqrt{\Delta \tau } \varepsilon _i(k) + \delta \Delta \nu _i(k\Delta \tau ) \zeta _i(k). \end{aligned} \end{aligned}$$

(16)

We consider that informed fish (resp. startled fish) independently share common heading angle between each subgroup individuals such that, in the following development, we consider group with a single informed fish and a single startle response fish. Since the informed fish move independently from other fish and its heading angle is the reference state, stability of the local disagreement $\theta _i - \theta _0$ of the system can be studied for the $N-1$ other fish at the proximity of a constant heading angle $\theta _0$ characterizing the coordinated state and taken as the reference state.

In the proximity of a given ordered state $\theta _0$, a mean square stability analysis over the system in (16) has been proposed in Mwaffo et al. (2015b), Mwaffo and Porfiri (2016) for the vectorial network model (VNM). We recall that the VNM model introduced in Aldana and Huepe (2003) is a simplification of the classical Vicsek model in the limit of larger speed where particles are considered to interact at any time instant with randomly selected particles in the group. This simplification is in particular true for small particle size where it is more likely that all particles interact with each other. In Mwaffo et al. (2015b), Mwaffo and Porfiri (2016), a linear approximation of the polarization order parameter is proposed and adapted for $N-1$ particles as:

$$\begin{aligned} \mathrm {Pol}(k) = \frac{1}{N-1} \sum _{i=1}^{N-1} \mathbf {v}_i= \frac{1}{N-1} \Big |\sum _{i=1}^{N-1} e^{\iota \theta _i(k)} \Big | \simeq 1-\frac{1}{2(N-1)}\rho (k), \end{aligned}$$

where $\mathbf {v}_i$ is the unit velocity vector, $\iota$ is the imaginary index, and $\rho (k) = \mathbf {E}\left[ \sum _{i=1}^{N-1} \left( \theta _i(k) - \theta _0\right) ^2\right]$ is the steady state deviation from the common synchronized state (see, for example, Mwaffo and Porfiri 2015b). For small noise values, when $0<\left( 1-\Delta \tau \right) ^2 < 1$, the steady-state deviation $\rho$ has been shown in (Mwaffo et al. 2015b; Mwaffo and Porfiri 2016) to converge toward a finite value. This yields after identification to a closed-form expression of the polarization as:

$$\begin{aligned} \mathrm {Pol} \simeq 1 - \frac{(\varsigma ^2\Delta \tau + \lambda \gamma ^2)}{2}\frac{N-2}{N-1} = 1-\frac{({\varsigma ^2}\Delta \tau +\iota \delta ^2)}{2}\frac{N-2}{N-1} \end{aligned}$$

(17)

where the number of connected neighbors is set to $|\mathcal {N}_i| = N-1$.

1.3 Group Response Evaluated with the Combined Parameter $P \wedge C$

See Fig. 11.

Fig. 11

Group coordination measured by the combined parameter $P \wedge C$ with respect to the informed fish (a) and with respect to the startle response fish (b) by varying the jump frequency $\iota$ and the jump intensity $\delta$. In the color maps, the alignment index P and the cohesion index C are identical to the parameters estimated in Fig. 7

1.4 Group Formation Pattern After Escape Response

See Fig. 12.

Fig. 12

Typical formation patterns observed after escape response for a smaller frequencies of the jump ($\iota = 0.022$ and $\delta =0.334$) resulting in fish aligning their heading in a direction prescribed by the startle fish and for b larger frequencies of the jump ($\iota = 0.0696$ and $\delta =0.334$) resulting in a circular motion. Other parameters are reported in Table 1