Nonlinear Control in the Nematode C. elegans

Abstract

Recent whole-brain calcium imaging recordings of the nematode C. elegans have demonstrated that the neural activity associated with behavior is dominated by dynamics on a low-dimensional manifold that can be clustered according to behavioral states. Previous models of C. elegans dynamics have either been linear models, which cannot support the existence of multiple fixed points in the system, or Markov-switching models, which do not describe how control signals in C. elegans neural dynamics can produce switches between stable states. It remains unclear how a network of neurons can produce fast and slow timescale dynamics that control transitions between stable states in a single model. We propose a global, nonlinear control model which is minimally parameterized and captures the state transitions described by Markov-switching models with a single dynamical system. The model is fit by reproducing the timeseries of the dominant PCA mode in the calcium imaging data. Long and short time-scale changes in transition statistics can be characterized via changes in a single parameter in the control model. Some of these macro-scale transitions have experimental correlates to single neuro-modulators that seem to act as biological controls, allowing this model to generate testable hypotheses about the effect of these neuro-modulators on the global dynamics. The theory provides an elegant characterization of control in the neuron population dynamics in C. elegans. Moreover, the mathematical structure of the nonlinear control framework provides a paradigm that can be generalized to more complex systems with an arbitrary number of behavioral states.

Keywords: C. elegans; dimensionality reduction; feed-forward control; neural network models; nonlinear control.

Figure 1

C. elegans neural activity in the PCA space of the first two modes. Trajectories colored by behavioral state.

Figure 2

(A) PCA activity of C. elegans 5. (B) Dynamical systems control model fit to PCA activity of C. elegans 5.

Figure 3

Timeseries of dominant mode of C. elegans neural activity [v₁(t)] and corresponding model variable [x(t)]. Models are fit to each C. elegans by minimizing the error [E(t)] between the PCA and model timeseries. Trajectories are colored by behavioral state.

Figure 4

(A–C) Phase plane, nonlinear stochastic activity, and state distributions of Equation (4) with increasing β values. (A) β = 0 generates equally stable fixed points. (B) β = 0.6 generates a less stable fixed point which turns into a slow point as the fixed points merge. (C) β, r₂ ∈ ℂ and the right fixed point is lost. (D) C. elegans PCA trajectory during a reversal bout and (E) the corresponding distribution. The forward fixed point is unstable during this interval. (F–H) C. elegans activity in a preferred 10% oxygen environment which promotes stability in the forward state compared with (I–K) C. elegans activity in an aversive 21% oxygen environment which destabilizes the forward state. (F,G) PCA activity and distribution of a single C. elegans in the preferred oxygen environment compared with the activity of this same C. elegans in the aversive oxygen environment (I,J). Average distribution for 10 C. elegans in the preferred environment (H) compared to the aversive environment (K).

Figure 5

Principal component analysis of neural activity. (A) Calcium imaging timeseries mean centered (time in seconds). (B) Timeseries of first principal component with moving average. (C) Timeseries of second principal component with moving average. (D) Neural activity in PCA space using uncorrected PCA. (E) Timeseries of first principal component with moving average subtracted. (F) Timeseries of second principal component with moving average subtracted. (G) Neural activity in PCA space using corrected PCA.

Figure 6

State distributions of nonlinear models for various parameter regimes. (A) Fixed point relative locations affects their stability. (B) Increasing levels of Brownian motion (σ) increases the variation about the fixed points. (C) More frequent control signals more evenly distributes the time spent in stable vs. transitional states. (D) Stronger damping in the system keeps trajectories close to fixed points.