Sparse linear regression

This protocol is extracted from research article:

The somatosensory cortex receives information about motor output

**
Sci Adv**,
Jul 10, 2019;
DOI:
10.1126/sciadv.aaw5388

The somatosensory cortex receives information about motor output

Procedure

We applied a Bayesian SLiR algorithm that introduces sparse conditions for the unit/channel dimension only and not for the temporal dimension of the model. High-γ power recorded in S1 electrodes was modeled as a weighted linear combination of the neuronal activity of peripheral afferents and high-γ activity in MCx using multidimensional linear regression as follows$${y}_{j,T}(t)={\displaystyle \sum _{k,l}}{w}_{j,k,l}\times {x}_{k,T}(t+\mathit{l}\delta )+{b}_{j}$$(1)where *y*_{j,T}(*t*) is a vector of activity of an S1 electrode *j* (two frequency bands of eight and six electrodes in Monkeys T and C, respectively) at time index *t* in a trial *T*, *x*_{k,T}(*t + l*δ) is an input vector of a peripheral afferent or a cortical electrode *k* at time index *t* and time lag *l*δ (δ = 5 ms) in a trial *T*, *w*_{j,k,l} is a vector of weights on a peripheral afferent or a cortical electrode *k* at time lag *l*δ, and *b _{j}* is a vector of bias terms to

To compute the contribution of each cortical area or peripheral afferents to the reconstruction of S1 activity, we calculated each component of reconstructed activity using MCx, premotor cortices, M1, or peripheral afferent activity and their respective weight values in a decoding model that was built from combined activities in MCx and peripheral afferents. For example, the MCx component was calculated as follows$$y\_{\text{MCx}}_{j,T}(t)={\displaystyle \sum _{k,l}}{w}_{j,k,l}\times x\_{\text{MCx}}_{k,T}(t+\mathit{l}\delta )+{b}_{j}$$(2)where *y_*MCx_{j,T}(*t*) is a vector of the MCx component at an S1 electrode *j* at time index *t* in a trial *T*, *x_*MCx_{k,T}(*t + l*δ) is an input vector of a cortical electrode *k* at time index *t* and time lag *l*δ in a trial *T*, and *w*_{j,k,l} is derived from a vector of weights in Eq. 1, but with weights assigned to peripheral afferents removed.

The temporal activity of muscles was modeled as a weighted linear combination of high-γ activity in M1 or S1 using the above Eq. 1. In the analysis, *y*_{j,T}(*t*) is a vector of EMG of a muscle *j* (12 and 10 muscles of Monkey T and C, respectively) at time index *t* in a trial *T*. *x*_{k,T}(*t + l*δ) is an input vector of a channel *k* at time index *t* and time lag *l*δ (δ = 5 ms) in a trial *T*. *w*_{j,k,l} is a vector of weights on a channel *k* at time lag *l*δ for a muscle *j*, and *b _{j}* is a vector of bias terms to

The initial peak EMG amplitude was modeled as a weighted linear combination of the high-γ activity in M1 or in S1 within an overlapping, sliding time window of 50 ms as follows$${y}_{j,T}={\displaystyle \sum _{k,l}}{w}_{j,k,l}\times {x}_{k,T}(t+\mathit{l}\delta )+{b}_{j}$$(3)where *y*_{j,T} is a vector of EMG of a muscle *j* (12 and 10 muscles of Monkey T and C, respectively) in a trial *T*, *x*_{k,T}(*t + l*δ) is an input vector of a channel *k* at time index *t* and time lag *l*δ in a trial *T*, *w*_{j,k,l} is a vector of weights on a channel *k* at time lag *l*δ for a muscle *j*, and *b _{j}* is a vector of bias terms to

Joint angles were modeled as a weighted linear combination of neuronal activities in peripheral afferents or high-γ power in M1 or S1 using multidimensional linear regression as follows$${y}_{j,T}(t)={\displaystyle \sum _{k,l}}{w}_{j,k,l}\times {x}_{k,T}(t+\mathit{l}\delta )+{b}_{j}$$(4)where *y*_{j,T}(*t*) is a vector of kinematic variables *j* (joint angle) at time index *t* in a trial *T*, *x*_{k,T}(*t + l*δ) is an input vector of unit *k* at time index *t* and time lag *l*δ (δ = 5 ms) in a trial *T*, *w*_{j,k,l} is a vector of weights on a peripheral afferent or a cortical electrode *k* at time lag *l*δ, and *b _{j}* is a vector of bias terms to

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