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Sensorimotor Transformations Maurice J. Chacron and Kathleen E. Cullen

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Outline Lecture 1: - Introduction to sensorimotor transformations - The case of “linear” sensorimotor transformations: refuge tracking in electric fish - introduction to linear systems identification techniques - Example of sensorimotor transformations: Vestibular processing, the vestibulo-occular reflex (VOR).

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Outline Lecture 2: - Nonlinear sensorimotor transformations - Static nonlinearities - Dynamic nonlinearities

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Lecture 1 Sensorimotor transformation: if we denote the sensory input as a vector S and the motor command as M, a sensorimotor transformation is a mapping from S to M : M =f(S) Where f is typically a nonlinear function

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Examples of sensorimotor transformations -Vestibulo-occular reflex -Reaching towards a visual target, etc…

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Example: Refuge tracking in weakly electric fish

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Refuge tracking

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Sensory input Motor output Error

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Results (Cowan and Fortune, 2007) -Tracking performance is best when the refuge moves slowly -Tracking performance degrades when the refuge moves at higher speeds -There is a linear relationship between sensory input and motor output

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Linear systems identification techniques

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Linear functions What is a linear function? So, a linear system must obey the following definition:

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Linear functions (continued) This implies the following: a stimulus at frequency f 1 can only cause a response at frequency f 1

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Linear transformations assume output is a convolution of the input with a kernel T(t) with additive noise. We’ll also assume that all terms are zero mean. -Convolution is the most general linear transformation that can be done to a signal

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An example of linear coding: Rate modulated Poisson process time time dependent firing rate

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Linear Coding: Example: Recording from a P-type Electroreceptor afferent. There is a linear relationship between Input and output Gussin et al. 2007 J. Neurophysiol.

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Instantaneous input-output transfer function:

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Fourier decomposition and transfer functions - Fourier Theorem: Any “smooth” signal can be decomposed as a sum of sinewaves -Since we are dealing with linear transformations, it is sufficient to understand the nature of linear transformations for a sinewave

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Linear transformations of a sinewave Scaling (i.e. multiplying by a non-zero constant) Shifting in time (i.e. adding a phase)

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Cross-Correlation Function For stationary processes: In general,

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Cross-Spectrum Fourier Transform of the Cross-correlation function Complex number in general a: real part b: imaginary part

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Representing the cross-spectrum: : amplitude : phase

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Transfer functions (Linear Systems Identification) assume output is a convolution of the input with a kernel T(t) with additive noise. We’ll also assume that all terms are zero mean. Transfer function

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Calculating the transfer function multiply by: and average over noise realizations =0

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Gain and phase:

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Sinusoidal stimulation at different frequencies Stimulus Response 20 msec

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Gain

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Combining transfer functions input output

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Where transfer functions fail…

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Vestibular system Cullen and Sadeghi, 2008

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Example: vestibular afferents CV=0.044CV=0.35

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` Regular afferent Firing rate (spk/s) Head velocity (deg/s) 120 100 80 60 40 20 0 -20 -40

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` Irregular afferent Firing rate (spk/s) Head velocity (deg/s) 160 140 120 100 80 60 40 -20 -40 20 0

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Signal-to-noise Ratio:

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Borst and Theunissen, 1999

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Using transfer functions to characterize and model refuge tracking in weakly electric fish Sensory input Motor output Error

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Characterizing the sensorimotor transformation 1 st order 2 nd order

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Modeling refuge tracking using transfer functions sensory input sensory processing motor processing motor output

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Modeling refuge tracking using transfer functions sensory input sensory processing motor output Newton

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Simulink demos

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Mechanics constrain neural processing

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Summary Some sensorimotor transformations can be described by linear systems identification techniques. These techniques have limits (i.e. they do not take variability into account) on top of assuming linearity.

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