Sensorimotor Transformations Maurice J. Chacron and Kathleen E. Cullen

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).

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

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

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

Example: Refuge tracking in weakly electric fish

Refuge tracking

Sensory input Motor output Error

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

Linear systems identification techniques

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

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

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

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

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

Instantaneous input-output transfer function:

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

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

Cross-Correlation Function For stationary processes: In general,

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

Representing the cross-spectrum: : amplitude : phase

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

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

Gain and phase:

Sinusoidal stimulation at different frequencies Stimulus Response 20 msec

Gain

Combining transfer functions input output

Where transfer functions fail…

Vestibular system Cullen and Sadeghi, 2008

Example: vestibular afferents CV=0.044CV=0.35

` Regular afferent Firing rate (spk/s) Head velocity (deg/s)

` Irregular afferent Firing rate (spk/s) Head velocity (deg/s)

Signal-to-noise Ratio:

Borst and Theunissen, 1999

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

Characterizing the sensorimotor transformation 1 st order 2 nd order

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

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

Simulink demos

Mechanics constrain neural processing

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.