Current-source density estimation based on inversion of electrostatic forward solution: Effects of finite extent of neuronal activity and conductivity discontinuities
Introduction
The technology for large-scale electrical recording of neural activity is rapidly improving, and this offers better opportunities for going beyond standard single-unit studies and measure the activity from populations of neurons Nadasdy et al., 1998, Buzsaki, 2004. Various types of multisite electrodes are used to simultaneously record firing of action potentials from numerous neurons, but new methods for data analysis are needed to fully exploit recordings of local field potentials from, for example, multishank laminar electrodes (Buzsaki, 2004).
Local field potentials, i.e., the low-frequency part of extracellularly recorded potentials, are thought to predominantly stem from dendritic processing of synaptic inputs, but a direct interpretation in terms of the underlying neural activity is difficult (Freeman and Nicholson, 1975). A standard measurement procedure has been to record the field potential at equidistant, linearly positioned electrode contacts using a laminar electrode vertically penetrating the cortical layers (see, e.g., Rappelsberger et al., 1981, Mitzdorf, 1985, Nadasdy et al., 1998, Ulbert et al., 2001). Under the assumptions of homogeneous cortical in-plane activity, constant extracellular electrical conductivity and equidistant electrode contacts, the current-source density (CSD) can be estimated from a double spatial derivative of the recorded potentials Freeman and Nicholson, 1975, Rappelsberger et al., 1981, Nakagawa and Matsumoto, 2000. This standard CSD estimation method only predicts the CSD at the interior electrode positions. Vaknin et al. (1988) suggested a procedure for also obtaining CSD estimates at the first and last electrode based on the assumption that the potential varies negligibly above the first and below the last electrode.
Cortical activity has a characteristic columnar organization Hubel and Wiesel, 1977, Mountcastle, 1997, and the in-plane homogeneity assumption is questionable. For the rat barrel cortex, where each whisker has a dedicated barrel column, Di and Barth (1991) found the surface potential measured above the barrels to be varying on the submillimeter scale following stimulation of the principal whisker. Nicholson and co-workers Nicholson and Llinas, 1971, Nicholson and Freeman, 1975, Freeman and Nicholson, 1975 studied cerebellar activity in various species and estimated the effective size of the columnar activity to be so small ( 0.5 mm; Nicholson and Llinas, 1971) that the applicability of standard CSD analysis was questioned. Indeed they recommended and later pursued a full three-dimensional CSD analysis where the potential was measured in all spatial directions (Nicholson and Llinas, 1975). The spatial extension of neural activity will in general depend on, for example, the type of cortex, type of stimulus (if stimulus-evoked responses are considered) and level and type of anesthesia.
In the present paper we develop a general method for estimation of CSDs from measured local field potentials where situations such as (i) spatially confined cortical activity and (ii) spatially varying extracellular conductivity can be handled. The method, labeled the i nverse CSD (iCSD) method, is based on the explicit inversion of the electrostatic forward solution. The calculation of the electrical potential from a given CSD distribution is in principle straightforward, and in our method this is exploited to solve the inverse problem, i.e., estimate the CSD distribution parameterized by N parameters from N measurements of the potential.
Preliminary results from this project have been presented earlier in poster format (Pettersen et al., 2004).
Section snippets
Matrix formulation of standard CSD method
A common starting point for the estimation of the current-source density from the extracellular field potential is Nicholson and Freeman, 1975, Mitzdorf, 1985where represents the electrical conductivity tensor which in general depends on position. The derivation of this relationship is based on the quasistatic approximation, and this approximation appears to be valid for the low temporal frequencies characteristic for the field potentials of interest Mitzdorf, 1985
Model-based test of iCSD method
We first performed a model study. Knowing the correct CSD allowed us to readily assess the accuracy of the methods; an option not available when using experimental data. In Fig. 2, the depth profile of the three model CSD distributions is shown. The first distribution consists of a region of constant positive current-source density (source) above a region of constant negative current-source density (sink). A similar highly idealized square-function distribution was also considered by Nicholson
Discussion
In the present paper we have introduced a new method for estimation of current-source density based on inversion of the electrostatic forward solution. The method, labeled the inverse CSD method, can be applied to data from various types of multielectrode geometries. Here, we have focused on linear-array (laminar) electrodes which is commonly used to probe cortical population activity. The standard method for CSD estimation from such potential recordings has been to apply a three-point discrete
Acknowledgements
We thank Martin Kermit for help with the data processing. This work was supported by the National Institute of Health Grants R01 EB00790 and NS18741 and by the Research Council of Norway.
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