The gradual and noisy accumulation of evidence is a fundamental component of decision-making, with noise playing a key role as the source of variability and errors. However, the origins of this noise have never been determined. We developed decision-making tasks in which sensory evidence is delivered in randomly timed pulses, and analyzed the resulting data with models that use the richly detailed information of each trial's pulse timing to distinguish between different decision-making mechanisms. This analysis allowed measurement of the magnitude of noise in the accumulator's memory, separately from noise associated with incoming sensory evidence. In our tasks, the accumulator's memory was noiseless, for both rats and humans. In contrast, the addition of new sensory evidence was the primary source of variability. We suggest our task and modeling approach as a powerful method for revealing internal properties of decision-making processes.

Gradual accumulation of evidence is thought to be fundamental for decision-making, and its neural correlates have been found in multiple brain regions1–8. Here we develop a generalizable method to measure tuning curves that specify the relationship between neural responses and mentally-accumulated evidence, and apply it to distinguish the encoding of decision variables in posterior parietal cortex (PPC) and prefrontal cortex (frontal orienting fields, FOF). We recorded the firing rates of neurons in PPC and FOF from rats performing a perceptual decision-making task. Classical analyses uncovered correlates of accumulating evidence, similar to previous observations in primates and also similar across the two regions. However, tuning curve assays revealed that while the PPC encodes a graded value of the accumulating evidence, the FOF has a more categorical encoding that indicates, throughout the trial, the decision provisionally favored by the evidence accumulated so far. Contrary to current views3,5,7–9, this suggests that premotor activity in frontal cortex does not play a role in the accumulation process, but instead has a more categorical function, such as transforming accumulated evidence into a discrete choice. To causally probe the role of FOF activity, we optogenetically silenced it during different timepoints of the trial. Consistent with a role in committing to a categorical choice at the end of the evidence accumulation process, but not consistent with a role during the accumulation itself, a behavioral effect was observed only when FOF silencing occurred at the end of the perceptual stimulus. Our results place important constraints on the circuit logic of brain regions involved in decision-making.

Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear.

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In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.

Numerous brain regions have been shown to have neural correlates of gradually accumulating evidence for decision-making, but the causal roles of these regions in decisions driven by accumulation of evidence have yet to be determined. Here, in rats performing an auditory evidence accumulation task, we inactivated the frontal orienting fields (FOF) and posterior parietal cortex (PPC), two rat cortical regions that have neural correlates of accumulating evidence and that have been proposed as central to decision-making. We used a detailed model of the decision process to analyze the effect of inactivations. Inactivation of the FOF induced substantial performance impairments that were quantitatively best described as an impairment in the output pathway of an evidence accumulator with a long integration time constant (>240 ms). In contrast, we found a minimal role for PPC in decisions guided by accumulating auditory evidence, even while finding a strong role for PPC in internally-guided decisions.DOI: http://dx.doi.org/10.7554/eLife.05457.001

The resulting analysis combines key features of performing PCA in space and power spectral analysis in time, making it particularly suitable for analyzing large-scale neural recordings.

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