Behavior-dLDS: A decomposed linear dynamical systems model for neural activity partially constrained by behavior

TL;DR

The ability of behavior-decomposed linear dynamical systems (b-dLDS) to decouple behavioral vs. internal computations on controlled, simulated data is demonstrated and improvements over a state-of-the-art model that uses behavior to supervise all dynamics based on behavior are shown.

Abstract

Brain-wide recordings of large-scale networks of neurons now provide an unprecedented view into how the brain drives behavior. However, brain activity contains both information directly related to behavior as well as the potential for many internal computations. Moreover, observable behavior is executed not only by the brain, but also by the spinal cord and peripheral nervous system. Behavior is a coarse-grained product of neural activity, and we thus take the view that it can be best represented by lower-dimensional latent neural dynamics. Capturing this indirect relationship while disambiguating behavior-generating networks from internal computations running in parallel requires new modeling approaches that can embody the parallel and distributed nature of large-scale neural populations. We thus present behavior-decomposed linear dynamical systems (b-dLDS) to disentangle simultaneously recorded subsystems and identify how the latent neural subsystems relate to behavior. We demonstrate the ability of b-dLDS to decouple behavioral vs. internal computations on controlled, simulated data, showing improvements over a state-of-the-art model that uses behavior to supervise all dynamics based on behavior. We then show that b-dLDS can further scale up to tens of thousands of neurons by applying our model to large-scale recording of a zebrafish hindbrain during the complex positional homeostasis behavior, wherein b-dLDS highlights behavior-related dynamic connectivity networks.

Figures (13)

• Figure 1: behavior-dLDS architecture.A: The latent, lower-dimensional neural manifold represents the processes that generate observable neural activity, mapped to the ambient space via $\bm{D}$, and behavior, via $\bm{\Psi}$. B: The established dLDS model, inside the dashed lines, is updated here with a mapping from dynamics coefficients $\bm{c}$ to behavior $\bm{b}$ via $\bm{\Psi}$. This contributes another regularization term to the joint inference of latent states $\bm{x}$ and dynamics coefficients $\bm{c}$.
• Figure 2: Two-independent-systems simulation with 10 behavior traces generated as scaled versions of the first ground truth dynamics coefficient. First trial (out of 50 randomly generated from the same dynamics operators) shown. $R^2$ calculated over all 50 samples. A,B: "Neural" ground truth and reconstruction by b-dLDS. Signal is generated from a set of 6 ground truth dynamics operators and corresponding dynamics coefficients, with $\bm{D}=\bm{I}$. C,D: Simulated behavior and reconstruction from b-dLDS, where behavior is simulated from the ground truth dynamics coefficients. E: True dynamics coefficients used to generate the data. F: Learned dynamics coefficients (absolute value). G: Learned dynamics operators (1-15). H: Learned $\bm{\Psi}$, where each column corresponds to a dynamics operator and dynamics coefficient. Only column 3 is nonzero, and it perfectly reconstructs column 1 of the ground truth $\bm{\Psi}$. This is the correct column to reconstruct because the first ground truth dynamics coefficient was used to generate the simulated behavior traces.
• Figure 3: b-dLDS vs. CLDS and PSID. CLDS learns only behavior-conditioned dynamics, while b-dLDS learns both behavior-generating and intrinsic ones. PSID learns only a shared latent neural-behavioral space, while b-dLDS learns a hierarchical relationship between latent neural states, latent dynamics, and behavior. (Note: CLDS requires an 80/20 train/test split, so all models were only trained/learned on 40 out of 50 simulated trials. Results shown are on the same 40 "train" trials across models.) A-G: b-dLDS, 2 independent systems simulation, 1 trials out of 40 shown, $\bm{D} = \bm{I}$, with simulated behavior generated as a linear combination of ground truth dynamics coefficients. $R^2$ calculated across 40 trials. A, B: 1 trace out of 8 shown, data and reconstruction. E: True dynamics coefficients $\bm{c}$. Stars match selected time points in H. Note that at the first two time points, one top-left (purple) dynamics operator and one bottom-right (yellow) dynamics operator are active, while at the third time point, only a top-left dynamics operator is active. D: b-dLDS-inferred $\bm{c}$. The b-dLDS model is initialized with more than enough dynamics operators (15 learned vs. 6 ground truth operators). Noisy dynamics operators in F correspond to unused (black) dynamics coefficients in D. E: Behavior and reconstruction (1 trace). G:$\bm{\Psi}$ is a row vector with one dynamics coefficient mapping to one behavior (true value: 1, learned value: 0.8765, other true values 0, learned near 0). H: CLDS learns an 8x8 dynamics matrix $\bm{A}_t$, while b-dLDS learns $\bm{f}$ and $\bm{c}$, which can be combined via $\sum_{m=1}^M f_m c_{mt}$ to create an equivalent matrix. We show 3 example time points here as described above. b-dLDS learns the block-diagonal dynamics matrices, while CLDS does not. The full comparison across time is shown in Figure \ref{['fig:CLDSAllTime']}. PSID learns a dynamics matrix $\bm{A}$ that describes the relationship between latent states. I: CLDS neural data reconstruction (1st trace). J: PSID neural data reconstruction (1st trace). K: b-dLDS achieves lower relative mean squared error than CLDS on dynamics reconstruction, regardless of the number of dynamics or the number of behaviors. CLDS used more than 1 TB of RAM when we tried to run it on 10 behavior traces.
• Figure 4: Four strongest $\bm{\Psi}$ connections between dynamics coefficients and motor activity in a zebrafish model with one behavior trace. Overall model $R^2 = 0.90$. Dynamic connectivity maps for dynamics operators and corresponding dynamics coefficients shown. Motor-aligned dynamics are utilized throughout the recording or a mix of earlier and/or later in the recording.
• Figure 5: Two-independent-systems simulation with 10 behaviors generated from the first two ground truth states, with some overlap in their combined effects.A: Ground truth signal fed into the model. B: Signal reconstruction from b-dLDS. C: True dynamics coefficients used to generate the data. D: Learned dynamics coefficients (absolute value). E: Learned dynamics operators (1-15). F: Learned $\bm{\Psi}$. G: Best reconstruction of the true $\bm{\Psi}$ (2 columns, other columns all 0). H: Highlighted dynamics coefficient used to simulate behavior data. I: Highlighted dynamics coefficient corresponding to best match of learned $\bm{\Psi}$. J: Highlighted corresponding dynamics operator. K: Simulated behavior ground truth. L: Behavior reconstruction.