Hybrid M-Sequence Method

An extension of the m-sequence technique for the analysis of multi-input nonlinear systems

Ethan A. Benardete and Jonathan D. Victor

In Advanced Methods of Physiological Systems Modeling, Volume III, ed. V. Z. Marmarelis. New York, Plenum. pp. 87-110 (1994)


White-noise analysis and related methods of nonlinear systems identification describe a physical system's response to its input in terms of "kernels" of progressively higher orders. A popular analytic scheme in the laboratory uses a class of pseudorandom binary sequences, m-sequences, as a test signal. The m-sequence method has several advantages for investigating linear and nonlinear systems: ease of implementation, rapid calculation of system kernels, and a solid theoretical framework. One difficulty with this method for nonlinear analysis comes from the algebraic structure of m-sequences: linear and nonlinear terms can be confounded, especially in the analysis of systems with many inputs.

We have developed a modification of the m-sequence method which allows control of these anomalies. This method is based on input signals consisting of a superposition of m-sequences whose lengths are relatively prime. The fast computational methods which facilitate kernel calculation for a single m-sequence input are readily extended to this new setting. We describe the theoretical foundation of this method and present an application to the study of ganglion cells of the macaque retina.
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The algorithm in a (large) nutshell

We consider estimation of the first- and second-order kernels for a system with two inputs.

The system is presented with signal s[1](t) at input 1, and signal s[2](t) at input 2. s[1](t) and s[2](t) are each derived from two m-sequences, m[1](t) and m[2](t), whose repeat periods M1 and M2 are relatively prime:
s[1](t) = m[1](t) + m[2](t)

s[2](t) = m[1](t+T1) + m[2](t+T2).

First-order kernels

To estimate the first-order kernels, cross-correlate the response R(t) against the m-sequence m[1](t):

C1(t) = < R(u)m[1](u-t) >.

An independent estimate of each of these kernels, h^2[1] and h^2[2], can be derived from the crosscorrelation of the response R(t) against the m-sequence m[2](t):
C2(t) = < R(u)m[2](u-t) >.

Second-order self- and cross-kernels

To estimate the second-order kernels, cross-correlate the response R(t) against the product of the underlying m-sequences m[1](t) and m[1](t):

C1,2(t[1],t[2]) = < R(u)m[1](u-t[1])m[2](u-t[2]) >.

As diagrammed below, the cross-correlation C1,2 contains estimates of the second-order kernels with respect to each input, as well as the second-order cross-kernel.
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Diagrammatic view of the algorithm for second-order kernels

This diagrams estimation of second-order kernels for a system with two inputs. Responses at eact time t are placed into the two dimensional array on the left, in a position corresponding to the position of signal with respect to its component m-sequences. That is, the response R(t) is positioned at (t1,t2), where
t1 = t (mod M1)
t2 = t (mod M2).

The cross-correlation
C1,2(t[1],t[2]) = < R(u)m[1](u-t[1])m[2](u-t[2]) >.

can be calculated by applying the fast m-transform sequentially along each axis. The transformed space (right panel) contains estimates of the second-order self- and cross-kernels.

The key to the algorithm is that because the m-sequence lengths M1 and M2 are relatively prime, all pairs of values of t1 and t2 occur exactly once, as t ranges from 0 to M1M2 - 1. This allows encoding of a function of two times (i.e., the second order-kernels) within a function of one time (the cross-correlation). It also allows us to "factor" the calculation of the cross-correlation into two stages.
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Technical notes

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Background reading on m-sequences

Victor, J.D. (1992) Nonlinear systems analysis in vision: overview of kernel methods. In Nonlinear Vision: Determination of Neural Receptive Fields, Function, and Networks, ed. R. Pinter and B. Nabet. Cleveland, CRC Press. pp. 1-37.
The m-sequence method and the inverse-repeat idea
Sutter, E.E. (1987) A practical nonstochastic approach to nonlinear time-domain analysis. In Advanced Methods in Physiological System Modelling. Vol.1. ed. V. Z. Marmarelis. Los Angeles, Univ. of Southern California. pp. 303-315.
The fast m-transform
Sutter, E.E. (1991) The fast m-transform: a fast computation of cross-correlation with binary m-sequences. SIAM J. Comput. 20, 686-694.
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