Nonstationary Stochastic Dynamics Underlie Spontaneous Transitions between Active and Inactive Behavioral States

Abstract

The neural basis of spontaneous movement generation is a fascinating open question. Long-term monitoring of fish, swimming freely in a constant sensory environment, has revealed a sequence of behavioral states that alternate randomly and spontaneously between periods of activity and inactivity. We show that key dynamical features of this sequence are captured by a 1-D diffusion process evolving in a nonlinear double well energy landscape, in which a slow variable modulates the relative depth of the wells. This combination of stochasticity, nonlinearity, and nonstationary forcing correctly captures the vastly different timescales of fluctuations observed in the data (∼1 to ∼1000 s), and yields long-tailed residence time distributions (RTDs) also consistent with the data. In fact, our model provides a simple mechanism for the emergence of long-tailed distributions in spontaneous animal behavior. We interpret the stochastic variable of this dynamical model as a decision-like variable that, upon reaching a threshold, triggers the transition between states. Our main finding is thus the identification of a threshold crossing process as the mechanism governing spontaneous movement initiation and termination, and to infer the presence of underlying nonstationary agents. Another important outcome of our work is a dimensionality reduction scheme that allows similar segments of data to be grouped together. This is done by first extracting geometrical features in the dataset and then applying principal component analysis over the feature space. Our study is novel in its ability to model nonstationary behavioral data over a wide range of timescales.

Keywords: behavioral state transitions; bistability; computer simulations; electric fish; spontaneous movement; stochastic differential equation.

Figure 1. The EODR forms a complex, bimodally distributed time series that is highly correlated with movement. Blue traces show the EODR for Fish A, while the red and green binary traces show the movement variable, processed by a transition detection scheme (see Materials and Methods, Transition detection). To allow comparison between individuals, the EODR has been rescaled as in Jun et al. (2014), making it unitless. Insets show three examples of active states and thus reveal the diversity of activity time courses in such states.

Figure 2.

Active states are characterized by five geometrical features, f₁ to f₅ (see Materials and Methods, Active state features) that allow the dimensionality of the EODR time series (blue trace) to be reduced. t_on and t_off are the movement onset and offset times, respectively.

Figure 3.

Correlations between features are revealed once the dataset is cast onto feature space. This makes PCA useful for segregating states of similar shape. A, 5-D scatter plot of the active state features from Fish A. Each point corresponds to a single active state. The size of the points represent feature 4, and the colours, feature 5. B, Same scatter as in A but projected onto the plane of the first two principle components. Again, each point (black and coloured) represents an active state, and colours are only added for visual representation of the different groups (colours are unrelated to those in A).

Figure 4.

The EODR time series is populated by a heterogeneous set of active state shapes. The left panels show active states that belong to the five groups identified in Figure 3B, all aligned with respect to the movement onset time (dashed vertical lines). The black traces are group averages. The groups of Fish B are extracted in exactly the same way as for Fish A.

Figure 5.

To account for key aspects of the data, we propose a dynamical system model consisting of a stochastic variable evolving in a bistable potential function that is modulated by a nonstationary, latent variable, s(t). When the tilt parameter, a(t), is zero, the potential function is symmetrical (dashed function). The potential function has two stable points: x = 1 represents the active state, and x = 0 the inactive state. For Fish A, not only does the potential function tilt back and forth, but the separation variable, d(t), is also modulated. For Fish B, the only source of nonstationarity is the tilt variable, a(t).

Figure 6.

The model captures the correlation (Fish A), or lack thereof (Fish B), between the transition amplitude (f₁, y-axes) and the duration of the active state (f₂, x-axes). Each black dot represents an active state, projected on the f₁ – f₂ plane. For visual reference, a least-square line is also shown (straight black lines). The linear correlation coefficient and the p value for each plot are also reported as r and p, respectively.

Figure 7.

Simulation results from model A (lower panel, blue trace) qualitatively match the data from Fish A (upper panel, blue trace). This segment of data are taken from the trace in Figure 1. Applying a moving average filter to the observed EODR yields the black trace, which is then used as the latent variable, s(t), for model A (see Materials and Methods, Estimation of the latent variable). The black arrows are examples of active states that are expected to be reproduced by the model, since they are longer than the averaging window used to obtain the s(t).

Figure 8.

Simulation results from model B (lower panel, blue trace) qualitatively match the data from Fish B (upper panel, blue trace). Applying a moving average filter to the observed EODR yields the black trace, which is then used as the latent variable for model B (see Materials and Methods, Estimation of the latent variable).

Figure 9.

The models for both fish capture the essential features of inactive-to-active transitions across different groups. Coloured traces represent the average of a given group, i.e., the traces from panel A and C correspond exactly to the black traces of Figure 4. All active states are aligned with respect to movement onset time, in the case of experimental data, or with respect to the time of an upward crossing of the model’s unstable point, in the case of simulation (vertical dashed lines).

Figure 10.

Group 1, from Fish A, shows a high propensity to return briefly to the active state shortly after the time of movement offset. This is superimposed on a slow, tail-like decay of the EODR. The gray traces in the upper panel correspond to the same active states that are shown in the upper left panel of Figure 4, but in this case, they are aligned with respect to the movement offset time, which allows a transition-triggered PDF to be compiled around movement offset (lower panel). Three representative traces in the upper panel are shown in color as examples.

Figure 11.

Top, Set of 30 Monte Carlo simulations of Equation 1, driven by a linear decay of the latent variable, s_local(t). The stable and unstable point of the system are shown in green and red, respectively, and the black stars are time stamps marking downward crossings of the unstable point. The system is initialized at x = 1 and allowed to stabilize before the potential landscape starts to tilt over. All the realisations in the upper panel obey the PDF shown in Figure 12. Bottom, All traces from the upper panel are aligned with respect to the time stamps, which then allows the transition-triggered PDF of Figure 13 to be obtained.

Figure 12.

The nonstationary solution of the Fokker-Planck equation associated with Equation 1, driven by a linear decay of the latent variable, s_local(t). The solution confirms the presence of a transient bistability period (dashed rectangle) that allows the system to briefly jump back to the active state immediately after a downward transition. The slope of s_local(t), i.e., the tilting rate, controls the duration of this bistability period, with a higher rate leading to a briefer period. This solution is obtained numerically with a custom partial differential equation solver using finite volume discretization and implicit time-stepping (see Results, Numerical integration of the Fokker-Planck equation). Traces from the upper panel of Figure 10 evolve according to this PDF.

Figure 13.

Monte Carlo simulations of the transiently bistable system yields a transition-triggered PDF that is qualitatively similar to that obtained from the data. The 30 iterations used to generate this distribution are first aligned with respect to the time when the unstable point of the system is crossed downward and are then processed in the same way as the experimental data to obtain the transition-triggered PDF shown here.

Figure 14. A, Inactive state RTDs. Models for both fish produce inactive state RTDs (coloured curves) that are consistent with those of the data, all of which are fitted by the stretched exponential distribution functions of Equation 6 (dashed curves). For comparison, gray curves show the best fit exponential densities for the RTDs. Fitting parameters for all sessions can be found in Table 2, for both experimental data and simulations. Shaded areas are 95% bootstrap confidence interval for the RTDs, obtained with Matlab’s “bootci” function (1000 bootstrap samples). The simulated RTDs shown here are obtained from a single realisation of Equation 1, driven by the s(t) associated with the appropriate recording session. The dotted vertical lines correspond to the value of the averaging window used to obtain s(t). B, Distribution of fitting parameters. These distributions are obtained from the ensemble of 100 iterations associated with the same sessions as in A. Dashed lines show the value of these parameters obtained from fitting the experimentally observed RTDs. These values correspond to the fourth and fifth columns of Table 2.