Introduction

4 Neural population dynamics during brain-computer interface

control

execution (Pandarinath et al., 2018b). Recurrent neural network models indicate that local recurrence could implement MC’s brief oscillations (Sussillo et al., 2015; Pandarinath et al., 2018b; Russo et al., 2018; O’Shea et al., 2022; Saxena et al., 2022), which provide a basis set for generating multiphasic muscle activity (Churchland et al., 2012; Sussillo et al., 2015).

Mathematically, a dynamical system can be described by a function 𝑓 mapping the system state 𝑥 and external inputs 𝑢 to the instantaneous change in state ^𝑑𝑥

𝑑𝑡 : 𝑑𝑥

𝑑𝑡 = 𝑓(𝑥(𝑡), 𝑢(𝑡))

Equation 4.1 where 𝑡 indicates time. Frequently (Shenoy et al., 2013; Sauerbrei et al., 2020), the system state is defined as the MC firing rate vector 𝒓, and 𝑓 is constrained to be an additive linear function ℎ that models local recurrence. The input 𝒖 can include external stimuli and the firing rates of other brain areas. In this simplification, Equation 4.1 can be written as:

𝑑𝒓

𝑑𝑡 = ℎ(𝒓(𝑡)) + 𝒖(𝑡).

Equation 4.2 The autonomous dynamical systems hypothesis posits that the system (MC) is self-contained during execution of prepared movements. That is, 𝒖 is negligible during movement execution, further simplifying Equation 4.2 into:

𝑑𝒓

𝑑𝑡 = 𝐴𝒓(𝑡) + 𝑏

Equation 4.3 where 𝐴 is the dynamics matrix and 𝑏 is an offset vector.

The autonomous dynamical systems hypothesis explained phenomena that had puzzled the RM framework. For example, neurons appear to change representational tuning with time, because they generate a temporal basis rather than directly driving the output (Shenoy et al., 2013;

Pandarinath et al., 2018a). And apparent modulation by multiple movement parameters was not literal parameter representation, but rather a basis set for downstream muscle read-out.

Three related claims follow from the autonomous dynamical systems hypothesis. First, because the local MC dynamics ℎ are stable (Gallego et al., 2020) and the external inputs 𝑢 are hypothesized to be negligible, generating different neural trajectories 𝒓(𝑡) is solely determined by specifying different initial conditions 𝒓(0) (Churchland et al., 2012; Pandarinath et al., 2018b; Vyas et al., 2020). Second, these dynamics cause movement; that is, neural perturbations disrupt movement if and only if the perturbation alters the task dynamics subspace (O’Shea et al., 2022).

Third, because external inputs like sensory feedback are hypothesized to be negligible, errors cannot be easily detected and then corrected, so pattern generation must be robust to noise. Noise-robust autonomous dynamical systems exhibit low tangling (described in MC by (Russo et al., 2018)) (i.e.,

smooth flow fields) and low divergence (described in supplementary motor area, but notably not in MC, by (Russo et al., 2020)).

An important open question is whether these same autonomous dynamical principles govern MC activity during brain-computer interface (BCI) control. BCIs aim to restore movement to people with motor disabilities by decoding motor intent directly from neural activity (Hochberg et al., 2006, 2012; Collinger et al., 2013c; Aflalo et al., 2015; Gilja et al., 2015; Wodlinger et al., 2015; Bouton et al., 2016; Ajiboye et al., 2017). BCI cursor control has long been considered analogous to able- bodied reaching (Serruya et al., 2002; Taylor et al., 2002; Hochberg et al., 2006; Hwang et al., 2013;

Golub et al., 2016; Inoue et al., 2018). Given the success of aDSH in explaining MC activity during able-bodied reaching, this analogy suggests that aDSH-based modeling could improve human BCI control of assistive devices (Kao et al., 2015; Pandarinath et al., 2018a). Such modeling has even extended successfully to movements beyond arm reaching (Hall et al., 2014; Stavisky et al., 2019;

Kalidindi et al., 2021).

The applicability of the autonomous dynamical systems hypothesis is not a given, however.

The aDSH has received various critiques, ranging from other explanations for the observed data’s structure (Lebedev et al., 2019; Kalidindi et al., 2021) to aDSH’s lack of generalization to dexterous movements (Sauerbrei et al., 2020; Suresh et al., 2020) or stopping movements (Russo et al., 2020).

From the perspective of applying aDSH to BCI control, a core limitation is the assumption that feedback is negligible. Compared to the arm movements usually modeled by aDSH, current BCIs read out from a miniscule number of neurons, so BCI control is slower and less precise (Shanechi, 2016; Willett et al., 2017a) and prone to nonstationarities (Jarosiewicz et al., 2015). As a result, high- performance BCI control benefits from rapid real-time feedback (Gilja et al., 2012; Shanechi et al., 2016, 2017; Willett et al., 2018, 2019). During able-bodied movement, motor cortex reflects proprioceptive signals that are missing during BCI control (Stavisky et al., 2018). Taken together, behavioral dynamics and sensory inputs differ substantially between BCI control and able-bodied arm reaching.

Dynamical systems modeling promises to improve neural decoding performance (Kao et al., 2015; Pandarinath et al., 2018a; Gallego et al., 2020; Karpowicz et al., 2022). Given the difference in behavioral dynamics between able-bodied reaching and BCI control, an open question is how well the aDSH applies to BCI movements across different applications. In order to probe the relative contributions of autonomous dynamics, initial conditions, sensory inputs, and non-sensory inputs to MC activity, we adapted variants of the common BCI cursor tasks that dissociated the various components of a dynamical system. When the BCI participant attempted ballistic reaches analogous to previous aDSH studies, MC activity reproduced the previously described rotational dynamics.

When the BCI participant sustained cursor movement for longer durations, the initial conditions were similar, yet the neural trajectories paused along that of ballistic movements. This divergence in neural trajectories would have required inputs from outside MC, inconsistent with the autonomous dynamical systems hypothesis. These inputs to MC were present even in the absence of sensory feedback, indicating that MC receives inputs from other brain areas during relatively simple behavior. Although autonomous dynamics may exist during able-bodied movement, MC is not constrained by such dynamics when a task requires otherwise.