Evidence for frequency-dependent extracellular impedance from the transfer function between extracellular and intracellular potentials

Abstract

We examine the properties of the transfer function F _T  = V _m / V _LFP between the intracellular membrane potential (V _m ) and the local field potential (V _LFP) in cerebral cortex. We first show theoretically that, in the subthreshold regime, the frequency dependence of the extracellular medium and that of the membrane potential have a clear incidence on F _T . The calculation of F _T from experiments and the matching with theoretical expressions is possible for desynchronized states where individual current sources can be considered as independent. Using a mean-field approximation, we obtain a method to estimate the impedance of the extracellular medium without injecting currents. We examine the transfer function for bipolar (differential) LFPs and compare to simultaneous recordings of V _m and V _LFP during desynchronized states in rat barrel cortex in vivo. The experimentally derived F _T matches the one derived theoretically, only if one assumes that the impedance of the extracellular medium is frequency-dependent, and varies as $1/\sqrt{\omega}$ (Warburg impedance) for frequencies between 3 and 500 Hz. This constitutes indirect evidence that the extracellular medium is non-resistive, which has many possible consequences for modeling LFPs.

Appendices

Appendix A: Impedance for non-ideal membranes

In this section, we derive the expressions for the impedance of non-ideal membranes, which take into account that the membrane capacitance cannot be charged instantaneously (see Bédard and Destexhe 2008). Still within the linear regime and for a spherical source, we have:

$$\begin{array}{lll} I_r &=& \sum\limits_{i=1}^{N}g_i~(t,V_{m})( V_{m}(t)-E_i) \\ I_c &=& C_{m}\frac{d V_c}{dt} \\ I_{m} &=& I_r +I_c \\ V_{m} &=& V_c + R_{\rm MW}C_{m}\frac{d V_c}{dt}=V_c + \tau_{\rm MW}\frac{d V_c}{dt} \end{array} $$

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where all parameters have the same definition as in the main text, except for r _MW, which is the Maxwell-Wagner resistance which gives the non-ideal aspect of the membrane capacitance. The associated time constant, τ _MW, is also known as “Maxwell-Wagner time”.

In the linear regime, we have

$$\begin{array}{lll} \Delta I_r &=& \sum\limits_{i=1}^{N}g_i~({\left\langle{{V_{m}}}\right\rangle}|_t) \Delta V_{m} \\ \Delta I_c &=& C_{m}\frac{d \Delta V_c}{dt} \\ \Delta I_{m} &=& \Delta I_r +\Delta I_c \\ \Delta V_{m} &=& \Delta V_c + \tau_{\rm MW} \frac{d \Delta V_c}{dt} \end{array} $$

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Thus, in these conditions, the system of equations associated to the membrane is linear with time-independent coefficients.

By expressing the variation of current produced by the cell as a function of the variation of membrane voltage, in Fourier space, we obtain:

$$\begin{array}{lll} \Delta I_r(f) &=& G_{m} \Delta V_{m}(f) \\ \Delta I_c(f) &=& i\omega C_{m} \Delta V_c(f) \\ \Delta I_{m}(f) &=& \Delta I_r(f) +\Delta I_c(f) \\ \Delta V_{m}(f) &=& \Delta V_c(f) + i\omega \tau_{\rm MW}\Delta V_c(f) \end{array} $$

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where

$$ G_{m}=\sum\limits_{i=1}^{N}g_{i}. $$

It follows that the membrane impedance is given by:

$$ Z_{m}(f)=\frac{\Delta V_{m}(f)}{\Delta I(f)} = \frac{R_{m}}{1+i\frac{\omega \tau_{m}}{1+i\omega \tau_{\rm MW}}} \label{ZmN} $$

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where $R_{m} =\frac{1}{G_{m}}$. Note that if we set τ _MW = 0, we recover the same expressions for the impedance of ideal membranes, as considered in the main text.

Appendix B: Polynomial averaging algorithm for frequency-dependent signals

The polynomial averaging technique consists of fitting a polynomial to the cumulative distribution of the amplitude of the signal in frequency space. According to this procedure, one evaluates the difference between the data and model via a minimization problem (in the frequency space) as follows

$${{\rm argmin}}_{\tau_{m}, \alpha} ||y(f) - \hat{y}(f, \tau_{m}, \alpha)||, \quad f\in[3~{\rm Hz}, 500~{\rm Hz}], $$

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where y(f) represents the transfer function (after a Fourier transform or PSD has been applied). $\hat{y}(\cdot)$ represent the various model transfer functions (i.e. Warburg, resistive or capacitive) parameterized by τ _m and α. Since the theory we develop only explains changes in the slope of the mean of y(f), then to a first order approximation the above equation can be re-written in the following way

$$\begin{array}{lll} &&{\kern-6pt} {{\rm argmin}}_{\tau_{m}, \alpha} ||<y>(f) - \hat{y}(f, \tau_{m}, \alpha)||, \\ && f\in[3~{\rm Hz}, 500~{\rm Hz}] \end{array} $$

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where the operator < · > is the mean of the data in frequency domain. Any technique can be taken to evaluate the mean such as the moving average. However, the moving average does not completely remove the variance and is not general enough. Since Fourier transform of the transfer function or its PSD show a large variance we remove entirely this variance by employing the following polynomial algorithmic filter.

• 1. The Fourier transform of the signals is integrated relative to frequency:

  $$ G(f) =\int_{f_{\min}}^{f}F(f')df' $$

  (36)

  where f _min is the minimal frequency considered with f ≤ F _max , and F(f′) is the signal for which the mean function must be obtained. This integration gives a function of frequency which is very close to the integral of the mean function, which is true for the signals considered here.

• 2. To smooth the function G(f), a minimum variance fit is performed using a third-degree polynomial:

  $$ G^*(f)=A_3f^3 + A_2f^2+A_1f+A_0 + \mbox{O}(f^4) . $$

  (37)

  Note that higher-order polynomials can be used to improve accuracy. However, in our case we did not observe any gain for orders larger than three.

• 3. This polynomial was formally derived to find the expression of the mean function < F > (f)

  $$ <F>(f)=\frac{dG^*}{df} =3A_3f^2 + 2A_2f+A_1 ~ . $$

  (38)

Note that this algorithm is general and does not depend on any hypothesis concerning the stationarity of the signal because the average function is calculated in Fourier space.