Memory signatures in path curvature of self-avoidant model particles are revealed by time delayed self mutual information

Abstract

Emergent behavior in active systems is a complex byproduct of local, often pairwise, interactions. One such interaction is self-avoidance, which experimentally can arise as a response to self-generated environmental signals; such experiments have inspired non-Markovian mathematical models. In previous work, we set out to find ``hallmarks of self-avoidant memory" in a particle model for environmentally responsive swimming droplets. In our analysis, we found that transient self-trapping was a spatial hallmark of the particle's self-avoidant memory response. The self-trapping results from the combined effects of behaviors at multiple scales: random reorientations, which occur on the diffusion scale, and the self-avoidant memory response, which occurs on the ballistic (and longer) timescales. In this work, we use the path curvature as it encodes the self-trapping response to estimate an ``effective memory lifetime" by analyzing the decay of its time-delayed mutual information and subsequently determining the longevity of significant nonlinear correlations. This effective memory lifetime (EML) is longer in systems where the curvature is a product of both self-avoidance and random reorientations as compared to systems without self-avoidance.

Figures (6)

• Figure 1: (A) Five independent self-avoidant paths for comparison to (B) five independent active Brownian particle paths with matching velocity $V$ and reorientation timescale $\tau$ derived from fitting the velocity correlation function shown in (C). Despite nearly identical VCFs, the paths are qualitatively different.
• Figure 2: Illustration of the straightness index computation for sample noisy data (dashed gray path). The arclength is estimated using distances at the granularity size ($g = 5$ in this example -- blue segments) while the beeline distance is estimated at the moving window size ($w=25$ in this example -- red segments). The SI for each moving window is the ratio of its beeline to its arclength distances.
• Figure 3: Estimation of nonlinear dynamical correlations in the mutual information between independently generated straightness indices across an ensemble to determine a sufficient separation window $W$ for use on a single time series to suppress these dynamical correlations. After some acclimation time $t_a$, we calculate the ensemble-sampled mutual information $MI(S(t_a); S(t_a + T))$, where $S(t_a)$ and $S(t_a + T)$ are the set of all straightness index values of each independently generated path at time $t_a$ and $t_a+T$. (We choose $t_a = 15$s.) We select $W = T$ satisfying $MI(S(t_a); S(t_a + T)) \approx 0$ as the appropriate separation window size for future use in time-sampled mutual information computations.
• Figure 4: First column: Three 60s long paths from the same generating parameters ($\mu=0.01$ and $V=6$) that yield varying levels of self-avoidant memory expression. Second column: Corresponding straightness index time series $S(t)$ for each of the three paths. Third column: The mutual information decay curves corresponding to window sizes $W = 1, 2, 4$. As expected, the curves only appear to decay convincingly to zero in the case that $W=4$.
• Figure 5: (A) The average mutual information as a function of delay time decreases as the value of $\mu$ increases and the self-avoidant memory response is suppressed. As these curves decrease in value, they cross significance thresholds (horizontal straight lines) at earlier times. (B) For each significance threshold, the effective memory lifetime (first crossing time) of self-avoidant paths (solid lines) is recorded as a function of $\mu$ and compared to ABP effective memory lifetimes (dashed lines) with matching $V$ and $\tau$.