Pre-processing and transfer entropy measures in motor neurons controlling limb movements

Abstract

Directed information transfer measures are increasingly being employed in modeling neural system behavior due to their model-free approach, applicability to nonlinear and stochastic signals, and the potential to integrate repetitions of an experiment. Intracellular physiological recordings of graded synaptic potentials provide a number of additional challenges compared to spike signals due to non-stationary behaviour generated through extrinsic processes. We therefore propose a method to overcome this difficulty by using a preprocessing step based on Singular Spectrum Analysis (SSA) to remove nonlinear trends and discontinuities. We apply the method to intracellular recordings of synaptic responses of identified motor neurons evoked by stimulation of a proprioceptor that monitors limb position in leg of the desert locust. We then apply normalized delayed transfer entropy measures to neural responses evoked by displacements of the proprioceptor, the femoral chordotonal organ, that contains sensory neurones that monitor movements about the femoral-tibial joint. We then determine the consistency of responses within an individual recording of an identified motor neuron in a single animal, between repetitions of the same experiment in an identified motor neurons in the same animal and in repetitions of the same experiment from the same identified motor neuron in different animals. We found that delayed transfer entropy measures were consistent for a given identified neuron within and between animals and that they predict neural connectivity for the fast extensor tibiae motor neuron.

Appendix A: Singular spectrum analysis

The SSA algorithm is described as follows (Golyandina and Zhigljavsky 2013): consider a time-series Z _T  = (z ₁,  … , z _T ). The embedding matrix of Z is obtained as:

$$ \boldsymbol{Z}={\left({z}_{ij}\right)}_{i,j=1}^{L,K}=\left(\begin{array}{ccccc}{z}_1& {z}_2& {z}_3& \dots & {z}_K\\ {}{z}_2& {z}_3& {z}_4& \dots & {z}_{K+1}\\ {}\vdots & \vdots & \vdots & \vdots & \vdots \\ {}{z}_L& {z}_{L+1}& {z}_{L+2}& \dots & {z}_T\end{array}\right)\kern3em $$

(5)

where K = T − L + 1 and L are window lengths considered for calculation (L ≤ T/2). It is also important to note that Z is a Hankel matrix and has equal elements on the anti-diagonals. A singular value decomposition (SVD) value is be applied to the matrix ZZ ^′, representing it as a sum of rank-one bi-orthogonal elementary matrices. We then denote λ _1, λ _2, …  , λ _L is as the eigenvalues of ZZ ^′ in decreasing order of magnitude λ ₁ ≥  …  ≥ λ _L  ≥ 0 and P = (P ₁, P ₂,  … , P _L ) is the orthonormal system of the eigenvectors of ZZ ^′ corresponding to these eigenvalues (Golyandina and Zhigljavsky 2013). We also define d as d = max(i, such that λ _i  > 0) = rank Z (in real data, we usually have d = min {L, K}).

The principal components (PCs) V _i (i = 1 ,  …  , d) of the embedding matrix are then obtained by ${V}_i={\boldsymbol{Z}}^{\prime }{P}_i/\sqrt{\lambda_i}$, and thus the trajectory matrix can be written as Z = Z ₁ + Z ₂ + Z ₃ +  …  + Z _d , where ${\boldsymbol{Z}}_i=\sqrt{\lambda_i}{P}_i{V_i}^{\prime}\left(i=1,\dots, d\right)$. These matrices have rank 1, and therefore are called elementary matrices (Hassani 2007).

The signal, then, can be reconstructed by selecting PCs according to their desired properties and then projecting them back to the original coordinates of the time-series. This is done by selecting and partitioning the indices i = 1 ,  …  , d into disjoint subsets I ₁ , I ₂ ,  …  , I _m . Then, for a given subset I = {i ₁, i ₂,  … , i _Q } the corresponding resultant matrix Z _I is defined as ${\boldsymbol{Z}}_I={\boldsymbol{Z}}_{i_1}+{\boldsymbol{Z}}_{i_2}+\dots +{\boldsymbol{Z}}_{i_Q}$ . This also leads to the SVD decomposition being represented as $\boldsymbol{Z}={\boldsymbol{Z}}_{I_1}+{\boldsymbol{Z}}_{I_2}+\dots +{\boldsymbol{Z}}_{I_m}$.

Diagonal averaging is then applied to a matrix ${X}_{I_k}$ producing the reconstructed time-series ${\overset{\sim }{\mathrm{Z}}}^{(k)}=\left({{\overset{\sim }{z}}_1}^{(k)},{{\overset{\sim }{z}}_2}^{(k)},\dots, {{\overset{\sim }{z}}_T}^{(k)}\right)$. In this way, the original series is decomposed into a sum of m reconstructed subseries:

$$ {z}_n=\sum \limits_{k=1}^m{\overset{\sim }{z}}_n^{(k)},\left(n=1,2,\dots, T\right)\kern2.75em $$

(6)

With this, SSA can be used as a tool for time-series smoothing, extraction of trends and extraction of oscillatory components (Golyandina and Zhigljavsky 2013).