Surya Ganguli

Memory traces in dynamical systems

To perform nontrivial, real-time computations on a sensory input stream, biological systems must retain a short-term memory trace of their recent inputs. It has been proposed that generic high-dimensional dynamical systems could retain a memory trace for past inputs in their current state. This raises important questions about the fundamental limits of such memory traces and the properties required of dynamical systems to achieve these limits. We address these issues by applying Fisher information theory to dynamical systems driven by time-dependent signals corrupted by noise. We introduce the Fisher Memory Curve (FMC) as a measure of the signal-to-noise ratio (SNR) embedded in the dynamical state relative to the input SNR. The integrated FMC indicates the total memory capacity. We apply this theory to linear neuronal networks and show that the capacity of networks with normal connectivity matrices is exactly 1 and that of any network of N neurons is, at most, N. A nonnormal network achieving this bound is subject to stringent design constraints: It must have a hidden feedforward architecture that superlinearly amplifies its input for a time of order N, and the input connectivity must optimally match this architecture. The memory capacity of networks subject to saturating nonlinearities is further limited, and cannot exceed N. This limit can be realized by feedforward structures with divergent fan out that distributes the signal across neurons, thereby avoiding saturation. We illustrate the generality of the theory by showing that memory in fluid systems can be sustained by transient nonnormal amplification due to convective instability or the onset of turbulence.

Compressed sensing, sparsity, and dimensionality in neuronal information processing and data analysis.

The curse of dimensionality poses severe challenges to both technical and conceptual progress in neuroscience. In particular, it plagues our ability to acquire, process, and model high-dimensional data sets. Moreover, neural systems must cope with the challenge of processing data in high dimensions to learn and operate successfully within a complex world. We review recent mathematical advances that provide ways to combat dimensionality in specific situations. These advances shed light on two dual questions in neuroscience. First, how can we as neuroscientists rapidly acquire high-dimensional data from the brain and subsequently extract meaningful models from limited amounts of these data? And second, how do brains themselves process information in their intrinsically high-dimensional patterns of neural activity as well as learn meaningful, generalizable models of the external world from limited experience?

Holographic protection of chronology in universes of the Godel type

We analyze the structure of supersymmetric Godel-like cosmological solutions of string theory. Just as the original four-dimensional Godel universe, these solutions represent rotating, topologically trivial cosmologies with a homogeneous metric and closed timelike curves. First we focus on phenomenological aspects of holography, and identify the preferred holographic screens associated with inertial comoving observers in Godel universes. We find that holography can serve as a chronology protection agency: The closed timelike curves are either hidden behind the holographic screen, or broken by it into causal pieces. In fact, holography in Godel universes has many features in common with de Sitter space, suggesting that Godel universes could represent a supersymmetric laboratory for addressing the conceptual puzzles of de Sitter holography. Then we initiate the investigation of microscopic aspects of holography of Godel universes in string theory. We show that Godel universes are T-dual to pp-waves, and use this fact to generate new Godel-like solutions of string and M-theory by T-dualizing known supersymmetric pp-wave solutions.

One-Dimensional Dynamics of Attention and Decision Making in LIP

Where we allocate our visual spatial attention depends upon a continual competition between internally generated goals and external distractions. Recently it was shown that single neurons in the macaque lateral intraparietal area (LIP) can predict the amount of time a distractor can shift the locus of spatial attention away from a goal. We propose that this remarkable dynamical correspondence between single neurons and attention can be explained by a network model in which generically high-dimensional firing-rate vectors rapidly decay to a single mode. We find direct experimental evidence for this model, not only in the original attentional task, but also in a very different task involving perceptual decision making. These results confirm a theoretical prediction that slowly varying activity patterns are proportional to spontaneous activity, pose constraints on models of persistent activity, and suggest a network mechanism for the emergence of robust behavioral timing from heterogeneous neuronal populations.

Environmental Boundaries as an Error Correction Mechanism for Grid Cells

Medial entorhinal grid cells fire in periodic, hexagonally patterned locations and are proposed to support path-integration-based navigation. The recursive nature of path integration results in accumulating error and, without a corrective mechanism, a breakdown in the calculation of location. The observed long-term stability of grid patterns necessitates that the system either performs highly precise internal path integration or implements an external landmark-based error correction mechanism. To distinguish these possibilities, we examined grid cells in behaving rodents as they made long trajectories across an open arena. We found that error accumulates relative to time and distance traveled since the animal last encountered a boundary. This error reflects coherent drift in the grid pattern. Further, interactions with boundaries yield direction-dependent error correction, suggesting that border cells serve as a neural substrate for error correction. These observations, combined with simulations of an attractor network grid cell model, demonstrate that landmarks are crucial to grid stability.

On simplicity and complexity in the brave new world of large-scale neuroscience

Technological advances have dramatically expanded our ability to probe multi-neuronal dynamics and connectivity in the brain. However, our ability to extract a simple conceptual understanding from complex data is increasingly hampered by the lack of theoretically principled data analytic procedures, as well as theoretical frameworks for how circuit connectivity and dynamics can conspire to generate emergent behavioral and cognitive functions. We review and outline potential avenues for progress, including new theories of high dimensional data analysis, the need to analyze complex artificial networks, and methods for analyzing entire spaces of circuit models, rather than one model at a time. Such interplay between experiments, data analysis and theory will be indispensable in catalyzing conceptual advances in the age of large-scale neuroscience.

Spatial Information Outflow from the Hippocampal Circuit: Distributed Spatial Coding and Phase Precession in the Subiculum

Hippocampal place cells convey spatial information through a combination of spatially selective firing and theta phase precession. The way in which this information influences regions like the subiculum that receive input from the hippocampus remains unclear. The subiculum receives direct inputs from area CA1 of the hippocampus and sends divergent output projections to many other parts of the brain, so we examined the firing patterns of rat subicular neurons. We found a substantial transformation in the subicular code for space from sparse to dense firing rate representations along a proximal-distal anatomical gradient: neurons in the proximal subiculum are more similar to canonical, sparsely firing hippocampal place cells, whereas neurons in the distal subiculum have higher firing rates and more distributed spatial firing patterns. Using information theory, we found that the more distributed spatial representation in the subiculum carries, on average, more information about spatial location and context than the sparse spatial representation in CA1. Remarkably, despite the disparate firing rate properties of subicular neurons, we found that neurons at all proximal-distal locations exhibit robust theta phase precession, with similar spiking oscillation frequencies as neurons in area CA1. Our findings suggest that the subiculum is specialized to compress sparse hippocampal spatial codes into highly informative distributed codes suitable for efficient communication to other brain regions. Moreover, despite this substantial compression, the subiculum maintains finer scale temporal properties that may allow it to participate in oscillatory phase coding and spike timing-dependent plasticity in coordination with other regions of the hippocampal circuit.

A Hebbian learning rule gives rise to mirror neurons and links them to control theoretic inverse models

Mirror neurons are neurons whose responses to the observation of a motor act resemble responses measured during production of that act. Computationally, mirror neurons have been viewed as evidence for the existence of internal inverse models. Such models, rooted within control theory, map-desired sensory targets onto the motor commands required to generate those targets. To jointly explore both the formation of mirrored responses and their functional contribution to inverse models, we develop a correlation-based theory of interactions between a sensory and a motor area. We show that a simple eligibility-weighted Hebbian learning rule, operating within a sensorimotor loop during motor explorations and stabilized by heterosynaptic competition, naturally gives rise to mirror neurons as well as control theoretic inverse models encoded in the synaptic weights from sensory to motor neurons. Crucially, we find that the correlational structure or stereotypy of the neural code underlying motor explorations determines the nature of the learned inverse model: random motor codes lead to causal inverses that map sensory activity patterns to their motor causes; such inverses are maximally useful, by allowing the imitation of arbitrary sensory target sequences. By contrast, stereotyped motor codes lead to less useful predictive inverses that map sensory activity to future motor actions. Our theory generalizes previous work on inverse models by showing that such models can be learned in a simple Hebbian framework without the need for error signals or backpropagation, and it makes new conceptual connections between the causal nature of inverse models, the statistical structure of motor variability, and the time-lag between sensory and motor responses of mirror neurons. Applied to bird song learning, our theory can account for puzzling aspects of the song system, including necessity of sensorimotor gating and selectivity of auditory responses to birds own song (BOS) stimuli.

Direction Selectivity in Drosophila Emerges from Preferred-Direction Enhancement and Null-Direction Suppression

Across animal phyla, motion vision relies on neurons that respond preferentially to stimuli moving in one, preferred direction over the opposite, null direction. In the elementary motion detector of Drosophila, direction selectivity emerges in two neuron types, T4 and T5, but the computational algorithm underlying this selectivity remains unknown. We find that the receptive fields of both T4 and T5 exhibit spatiotemporally offset light-preferring and dark-preferring subfields, each obliquely oriented in spacetime. In a linear-nonlinear modeling framework, the spatiotemporal organization of the T5 receptive field predicts the activity of T5 in response to motion stimuli. These findings demonstrate that direction selectivity emerges from the enhancement of responses to motion in the preferred direction, as well as the suppression of responses to motion in the null direction. Thus, remarkably, T5 incorporates the essential algorithmic strategies used by the Hassenstein–Reichardt correlator and the Barlow–Levick detector. Our model for T5 also provides an algorithmic explanation for the selectivity of T5 for moving dark edges: our model captures all two- and three-point spacetime correlations relevant to motion in this stimulus class. More broadly, our findings reveal the contribution of input pathway visual processing, specifically center-surround, temporally biphasic receptive fields, to the generation of direction selectivity in T5. As the spatiotemporal receptive field of T5 in Drosophila is common to the simple cell in vertebrate visual cortex, our stimulus-response model of T5 will inform efforts in an experimentally tractable context to identify more detailed, mechanistic models of a prevalent computation. SIGNIFICANCE STATEMENT Feature selective neurons respond preferentially to astonishingly specific stimuli, providing the neurobiological basis for perception. Direction selectivity serves as a paradigmatic model of feature selectivity that has been examined in many species. While insect elementary motion detectors have served as premiere experimental models of direction selectivity for 60 years, the central question of their underlying algorithm remains unanswered. Using in vivo two-photon imaging of intracellular calcium signals, we measure the receptive fields of the first direction-selective cells in the Drosophila visual system, and define the algorithm used to compute the direction of motion. Computational modeling of these receptive fields predicts responses to motion and reveals how this circuit efficiently captures many useful correlations intrinsic to moving dark edges.

Evidence for a causal inverse model in an avian cortico-basal ganglia circuit.

Significance Auditory neural responses mirror motor activity in a songbird cortical area. The average temporal offset of mirrored responses is roughly equal to short sensorimotor loop delays. This correspondence between mirroring offsets and loop delays constitutes evidence for a causal inverse model. Causal inverse models can map a desired sensation into the required action. Learning by imitation is fundamental to both communication and social behavior and requires the conversion of complex, nonlinear sensory codes for perception into similarly complex motor codes for generating action. To understand the neural substrates underlying this conversion, we study sensorimotor transformations in songbird cortical output neurons of a basal-ganglia pathway involved in song learning. Despite the complexity of sensory and motor codes, we find a simple, temporally specific, causal correspondence between them. Sensory neural responses to song playback mirror motor-related activity recorded during singing, with a temporal offset of roughly 40 ms, in agreement with short feedback loop delays estimated using electrical and auditory stimulation. Such matching of mirroring offsets and loop delays is consistent with a recent Hebbian theory of motor learning and suggests that cortico-basal ganglia pathways could support motor control via causal inverse models that can invert the rich correspondence between motor exploration and sensory feedback.