Table 2 Equations to model HD-MEA signals under tissue slices and with saline cover19,37,59,72,73.
From: Accurate signal-source localization in brain slices by means of high-density microelectrode arrays
Model | Equation |
|---|---|
Bounded MoI, isotropic tissue, (z < hT) | Equation 4 \({{\rm{\varphi }}}_{{\rm{bo}},{\rm{z}} < {{\rm{h}}}_{{\rm{T}}}}({\rm{x}}^{\prime} ,{\rm{y}}^{\prime} ,0)={{\rm{\varphi }}}_{{\rm{se}},{{\rm{\sigma }}}_{{\rm{T}}}}({\rm{x}}^{\prime} ,{\rm{y}}^{\prime} ,0)+{{\rm{\gamma }}}_{{\rm{TS}}}+{\rm{C}}\) where \({{\rm{\gamma }}}_{{\rm{TS}}}=2\sum _{{\rm{n}}=1}^{\infty }\,{{\rm{W}}}_{{\rm{TS}}}^{{\rm{n}}}\,[{{\rm{\varphi }}}_{{{\rm{\sigma }}}_{{\rm{T}}}}({\rm{x}}^{\prime} ,{\rm{y}}^{\prime} ,-\,2{{\rm{nh}}}_{{\rm{T}}})+{{\rm{\varphi }}}_{{{\rm{\sigma }}}_{{\rm{T}}}}({\rm{x}}^{\prime} ,{\rm{y}}^{\prime} ,2{{\rm{nh}}}_{{\rm{T}}})]\), WTS = (σT − σS)/(σT + σS), σT is the conductivity of the acute tissue, σS corresponds to the saline cover and hT is the thickness of the tissue. The conductivity used in \({{\rm{\varphi }}}_{{\rm{se}}}\) is σT. C is the offset due to the height hS of the saline cover. |
Semi-infinite MoI, anisotropic tissue \({{\rm{\sigma }}}_{{{\rm{T}}}_{{\rm{a}}}}\) | Equation 5 \({{\rm{\varphi }}}_{{\rm{se}},{{\rm{\sigma }}}_{{{\rm{T}}}_{{\rm{a}}}}}({\rm{x}}^{\prime} ,{\rm{y}}^{\prime} ,0)=\frac{{\rm{I}}}{2{\rm{\pi }}\sqrt{{{\rm{\sigma }}}_{{\rm{Ty}}}{{\rm{\sigma }}}_{{\rm{Tz}}}{({\rm{x}}^{\prime} -{\rm{x}})}^{2}+{{\rm{\sigma }}}_{{\rm{Tx}}}{{\rm{\sigma }}}_{{\rm{Tz}}}{({\rm{y}}^{\prime} -{\rm{y}})}^{2}+{{\rm{\sigma }}}_{{\rm{Tx}}}{{\rm{\sigma }}}_{{\rm{Ty}}}{({\rm{z}}^{\prime} -{\rm{z}})}^{2}}}\) |
Bounded MoI, anisotropic tissue \({{\rm{\sigma }}}_{{{\rm{T}}}_{{\rm{a}}}}\), (z < hT) | Equation 6 \({{\rm{\varphi }}}_{{\rm{bo}},{\rm{z}}^{\prime} < {\rm{h}}}({\rm{x}}^{\prime} ,{\rm{y}}^{\prime} ,0)={{\rm{\varphi }}}_{{\rm{se}},{{\rm{\sigma }}}_{{{\rm{T}}}_{{\rm{a}}}}}({\rm{x}}^{\prime} ,{\rm{y}}^{\prime} ,0)+{{\rm{\gamma }}}_{{{\rm{T}}}_{{\rm{a}}}}+{\rm{C}}\) where \({{\rm{W}}}_{{{\rm{T}}}_{{\rm{a}}}{\rm{S}}}\equiv \frac{\sqrt{{{\rm{\sigma }}}_{{\rm{Tx}}}{{\rm{\sigma }}}_{{\rm{Ty}}}}-\sqrt{{{\rm{\sigma }}}_{{\rm{Sx}}}{{\rm{\sigma }}}_{{\rm{Sy}}}}}{\sqrt{{{\rm{\sigma }}}_{{\rm{Tx}}}{{\rm{\sigma }}}_{{\rm{Ty}}}}+\sqrt{{{\rm{\sigma }}}_{{\rm{Sx}}}{{\rm{\sigma }}}_{{\rm{Sy}}}}}\) assuming that both tissue and saline have similar planar anisotropy, i.e., σTx/σTy = σSx/σSy. |
