Journal of Computational Neuroscience

, Volume 29, Issue 3, pp 389–403 | Cite as

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

Intracellular-LFP transfer function
  • Claude Bédard
  • Serafim Rodrigues
  • Noah Roy
  • Diego Contreras
  • Alain Destexhe
Article

Abstract

We examine the properties of the transfer function FT = Vm / VLFP between the intracellular membrane potential (Vm) and the local field potential (VLFP) 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 FT. The calculation of FT 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 Vm and VLFP during desynchronized states in rat barrel cortex in vivo. The experimentally derived FT 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.

Keywords

Computational models Local field potentials EEG Extracellular resistivity Intracellular recordings Maxwell equations 

1 Introduction

There is a widespread consensus that mechanisms for generating the intracellular electrical activity are very well understood, however not complete. In contrast, much less is known about the the genesis of extracellular potentials, which is a subject of intense research. This is associated to the difficulty in assigning measurements of the extracellular potentials to a unique neurophysiological generator, which makes modeling of LFP/EEG a complex issue. Some of these mechanisms are believed to be related to synaptic activity, synchronous population spikes, ephaptic interactions, ionic dynamics, morphological structure of the neurons and many other processes (reviewed in Jefferys 1995; Nunez and Srinivasan 2006).

One of the characteristics of extracellular potentials is the very steep attenuation of “fast” events such as spikes, which are visible only within the immediate vicinity (a few microns) of the electrode. In contrast, “slow” events such as synaptic potentials are visible for much larger distances, typically a few hundred microns (Destexhe et al. 1999; Katzner et al. 2009). One way to explain this differential filtering is that the extracellular medium acts as a powerful low-pass filter (Bédard et al. 2004, 2006a, b). However, this aspect is controversial because some measurements of brain conductivity did not display significant filtering effects (Logothetis et al. 2007) while other measurements did (Ranck 1963; Gabriel et al. 1996). Some of these experiments (Gabriel et al. 1996) used careful controls, such as correcting for electrode polarization, showing the different frequency-dependence of various biological tissues, but most importantly, the independent measurements of conductivity and permittivity with theoretical constraints (Kramers-Kronig relations) (Gabriel et al. 1996). The recent measurements employed a sophisticated four-electrode measurement setup to yield more accurate measures (Logothetis et al. 2007). However, despite such controls, these measurements used current intensities that are much larger than biological sources, which may explain the discrepancy (Bédard and Destexhe 2009).

In the present paper, we provide theoretical work and analyze experimental measurements to examine whether the extracellular medium is non-resistive. We examine a quantity which depends on the extracellular impedance, the transfer function between simultaneously recorded intracellular and extracellular potentials. We show theoretically that, in the linear regime and for desynchronized states, the transfer function calculated from an (intracellular) single recording site strongly depends on the extracellular impedance, and can therefore be used to investigate its frequency dependence. We show preliminary results from desynchronized states in vivo, which indicate that the extracellular medium is indeed frequency dependent. We relate these findings to previous work and discuss their possible implications for modeling LFPs/EEG.

2 Methods

The experimental data used in this paper were taken from a large database of cells (Wilent and Contreras 2005a, b), in which we selected intracellular recordings with long periods of subthreshold activity, simultaneous with LFP recordings in the vicinity (1 mm) of the intracellular electrode, and in light anesthesia with desynchronized EEG.

2.1 Surgery and preparation

Experiments were conducted in accordance with the ethical guidelines of the National Institutes of Health and with the approval of the Institutional Animal Care and Use Committee of the University of Pennsylvania. Adult male Sprague–Dawley rats (300–350 g, n = 35) were anesthetized with isoflurane (5% for induction, 2% during surgery), paralyzed with gallamine triethiodide, and artificially ventilated. End tidal CO2 (3.5–3.7%) and heart rate were continuously monitored. Body temperature was maintained at 37 °C via servo-controlled heating blanket and rectal thermometer (Harvard Apparatus, Holliston, MA). The rat was placed in a stereotaxic apparatus and a craniotomy was made directly above the barrel cortex (1.0–3.0 mm A/P, 4.0–7.0 mm M/L), and the dura was resected. The cisterna magna was drained to improve stability. For intracellular recordings, additional measures were taken to improve stability, including dexamethasone (10 mg/kg, i.p.) to reduce brain swelling, hip suspension, and filling the craniotomy with a solution of 4% agar.

2.2 Electrophysiological recordings

Recordings of local field potentials (LFPs) across the cortical depth were performed with 16-channel silicon probes (Neuronexus, Ann Arbor, MI). Probe recording sites were separated by 100 \(\upmu\)m and had impedances of 1.5–2.0 MΩ at 1 kHz. The probe was lowered into the brain under visual guidance, oriented normal to the cortical surface, until the most superficial recording site was aligned with the surface. All neurons were regular-spiking cells, and had spikes with about 2 ms width, so they were presumably excitatory. All LFP signals shown here were obtained by pairs of closely-located (400–500 \(\upmu\)m apart) electrodes arranged vertically (surface-depth), and were amplified and band-pass filtered at 0.1–10 kHz (FHC, Inc., Bowdoinham, ME).

Intracellular recordings were performed in barrel cortex with glass micropipettes pulled on a P-97 Brown Flaming puller (Sutter Instrument Company, Novato, CA). Pipettes were filled with 3M potassium acetate and had DC resistances of 60–80 MΩ. A high-impedance amplifier (low-pass filter of 5 kHz) with active bridge circuitry (Neurodata, Cygnus Technology, Inc., Delaware Water Gap, PA) was used to record and inject current into cells. Vertical depth was measured by the scale on the micromanipulator. A Power 1401 data acquisition interface and Spike2 software (Cambridge Electronic Design, Cambridge, U.K.) were used for data acquisition.

2.3 Analysis and simulations

All simulations and analyses were realized using MATLAB (Mathworks Inc, Natick, MA), except for compartmental model simulations (see below).

The analysis of the PSD was performed using two different methods. A constrained nonlinear least square fit of the analytic expressions for different possible transfer functions (see Eqs. (27) and (22) in Sections 4 and 5) was performed to the transfer function calculated from experimental data. The fit was constrained to frequencies between 3 and 500 Hz and the parameters of the transfer functions were constrained to the physiological range (see Section 5 for details).

Because the PSD and calculated transfer functions had large variance, two different methods were used to perform the fitting. First, a polynomial averaging algorithm was used to calculate the mean value of the PSD, and the fit was performed against this mean value (see details in Appendix B). Second, a moving average window procedure which consisted of partitioning the data set into five epochs (corresponding to about 7.7 sec for one cell and about 4.4 sec for the second cell, and additionally applied a Hanning window to each epoch (a Hamming window was tested as well, and gave similar results). This method markedly reduces the variance of the PSD and of the transfer function. Other choices for window length were also tested and yielded similar results as those shown in the figures (not shown).

2.4 Compartmental model simulations

Compartmental models were simulated using the freely-available NEURON simulation environment (Hines and Carnevale 1997). A 16-compartment “ball-and-stick” model was simulated, based on a soma (area of 500 \(\upmu\)m2) and 15 dendritic compartments of equal length (46.6 \(\upmu\)m) and tapering diameter (from 4 \(\upmu\)m at the soma up to 1 \(\upmu\)m at the distal tip), so that the total cell area was of 6000 \(\upmu\)m2. Each compartment had passive properties, with axial resistivity of 250 Ω·cm, specific capacitance of 1 \(\upmu\)F/cm2, resting membrane conductance of 0.45 mS/cm2 and reversal potential of − 80 mV.

To simulate high-conductance states similar to in vivo measurements, each dendritic compartment contained two randomly fluctuating synaptic conductances (model from Destexhe et al. 2001) which were adjusted so that the mean excitatory and inhibitory conductances were respectively of 0.73 and 3.67 times the resting conductance. These values represent mean values of the “spontaneous” synaptic activity during activated states in vivo, according to intracellular measurements (see details in Destexhe et al. 2003). We verified that using these parameters, the subthreshold Vm activity of the cell was conform to typical intracellular measurements during desynchronized-EEG states, as for example in the experiments reported here.

In a separate set of simulations, the synaptic inputs were present only in the most distal compartment. In this case, all synaptic conductances were set to zero in all other compartments.

The LFP generated by this compartmental model was calculated at the level of the stem dendrite, 50 μm lateral to the dendritic axis, assuming a purely resistive medium, according to the expression:
$$ V_{\rm LFP} \ = \ {R_e \over 4 \pi} \ \sum\limits_{j} { I_j \over r_j }, $$
(1)
where Re = 230 Ω·cm is the extracellular resistivity (Ranck 1963), Ij is the total membrane current of compartment j, and rj is the distance between compartment j and the extracellular position where the LFP is evaluated. The LFP of the full model was compared to the LFPs calculated from each individual current source similar to above, with \(V_{\rm LFP}^{(j)} \ = \ {R_e \over 4 \pi} \ { I_j \over r_j }\).

3 Theory

In this section we formulate a theoretical relationship between extracellular and intracellular sub-threshold voltage activity. We consider a linear electromagnetic regime, which is defined by the fact that the linking equations are linear in frequency space. In such case we have jf = σfEf, Df = εfEf and Bf = μfHf where the second-order symmetric tensors σf, εf and μf do not depend on Ef and Hf. σf, εf and μf depend on position in general. In other words, we will work in the subthreshold voltage range, for which we assume linear current-voltage relations, and that conductances are not voltage dependent. Henceforth, we will refer to this dynamical state as ”linear regime”. We also assume that the cortical tissue is macroscopically isotropic, in which case the tensors become scalar functions.

We will also formulate the model in the frequency domain, to easily enable comparison between the amount of intracellular and extracellular signal within each frequency band, which is readily accessible from experimental data. The advantage of this approach is simplicity as it avoids tackling directly the different time scales inherently associated to LFP/EEG signals, many of which are not yet understood. For example, it is known that LFP/EEG signals are composed of mixed-mode oscillations, however the origins and transitions between these modes are not yet explained (Erchova and McGonigle 2008). The choice of working in the frequency domain also seems natural because the differential equations describing electric potentials transform into algebraic equations in frequency space.

The formulation of the relation between transmembrane (Vm) and extracellular (VLFP) activity requires to derive the transfer function, FT, that measures the ratio between the impedance of the cellular membrane and extracellular medium. To obtain this, we start from a general formalism where VLFP is expressed as a function of a large number of current sources located at different positions in extracellular space (Section 3.1). We then show under what conditions this formalism can reduce to a single current source, and derive the impedance for this current source (Section 3.3) and derive the corresponding transfer functions (Sections 3.4 and 3.5).

In the following, we use the notation F(f) or Ff to denote the function F in Fourier space with frequency f and ω = 2 πf.

3.1 General model with multiple current sources

We start from the general model in Fourier space:
$$ V_{\rm LFP}(r_i,f) = \sum\limits_{j=1}^{N} \frac{R_{j}}{d_{ij}} Z_j^{\rm med}(f) I_j(f) ~ , \label{geneq} $$
(2)
where VLFP(ri,f) is the extracellular potential at a position ri as resulting from a set of N monopolar current sources Ij(f). Rj is a constant, dij is the distance between source j and position ri, and \(Z_j^{\rm med}\) is the impedance of the extracellular medium around source j. This model is based on the property that any charge distribution in space and frequency can be expressed as a sum of monopolar sources. This therefore applies to complex current distributions in the complex dendritic structures of many neurons surrounding the recording site ri (see Fig. 1(a)), and thus, this formalism is general.
Fig. 1

Arrangement of potentials and impedances. (a) Scheme of the general model where the local field potential (VLFP) is generated by a large number of individual monopolar current sources (red) distributed in soma and dendrites of neurons in the network. If the activity of these current sources is desynchronized, the system can reduce to a single current source using a mean-field approximation. (b) Scheme of an individual current source, which is assumed to be of spherical symmetry with homogeneous extracellular medium, so that there is no position-dependence of conductivity or permittivity. Vi and Ve are the electric potentials, respectively inside and outside of the membrane, relative to a reference potential \(R_f^{\infty}=0\) situated at an infinite distance. The membrane potential Vm is the difference between Vi and Ve. VLFP is the local field potential, which is the voltage difference between a point P in extracellular space and \(R_f^{\infty}\). (c) Equivalent electrical circuit for each current source, where Zm is the impedance of the membrane, while Za and Zb are impedances of the extracellular medium

If we assume that the extracellular medium is electrically homogeneous, the extracellular impedance \(Z_j^{\rm med}\) is the same for every pair of points i,j, and the local field potential becomes:
$$ V_{\rm LFP}(r_i,f) = Z^{\rm med}(f) \sum\limits_{j=1}^{N} \frac{R_{j}}{d_{ij}} I_j(f) \label{geneq2} $$
(3)
We also assume that the network activity is of low correlation, as typically found during desynchronized network states in vivo, such as in awake animals, or during the “up-states” of anesthesia. In such states, it was shown that the activity is very irregular with low levels of correlation between cortical neurons (Destexhe et al. 1999; Gawne and Richmond 1993; Steriade 2001; Zohary et al. 1994). As a consequence, we can consider that the dendritic structure is bombarded by noisy synaptic events that are essentially uncorrelated, and if we assume that this activity primes over the deterministic link between individual current sources (see Section 3.2 for a test of this assumption), then the field produced by the ensemble of N current sources is equivalent to the field produced by independent sources. the variables \(\frac{R_{j}}{d_{ij}}\) and Ij can be considered as statistically independent, and by averaging over the ensemble of current sources, we obtain:
$$\begin{array}{lll} V_{\rm LFP}(r_i,f) &=& Z^{\rm med}(f) \sum\limits_{j=1}^{N} \frac{R_{j}}{d_{ij}} I_j(f)\\ &=& Z^{\rm med}(f) \ \bar{R}_D \ \bar{I}(f) ~. \end{array} $$
(4)
Here, the LFP has been expressed as a function of a “mean current source” (in the spatial sense) \(\bar{I}(f) = N <I_j(f)>|_j\) and where \(\bar{R}_D = <\frac{R_{j}}{d_{ij}}>|_j\).1 This formulation is equivalent to a mean-field theory expressed in Fourier space and where we consider the LFP as generated by the mean current source contribution.

3.2 Testing the assumptions of the formalism

To verify some of the assumptions of the formalism, we have simulated a ball-and-stick dendritic model subject to fluctuating synaptic conductances (Fig. 2). In a first set of simulations, the synaptic conductances (model from Destexhe et al. 2001) were located only at a single dendritic location. This case produced LFPs which were markedly filtered by the dendritic structure (Fig. 2(a), middle panel), as described previously (Pettersen and Einevoll 2008). This type of filtering is due to the fact that in dendritic structures, the return current (which participates to generate the LFP) is filtered by the membrane. In another set of simulations, we have considered a more realistic situation in which the synaptic conductances were located at all locations of the dendritic structure, so that the total conductance matches that measured experimentally in vivo (Destexhe et al. 2003). In this case, the filtering due to morphology was negligible (Fig. 2(b), middle panel), presumably because synaptic currents largely dominate over axial currents. These simulations suggest that the filtering due to morphology can be neglected for in vivo–like conditions, and that in such conditions, the filtering must come from another origin, such as the extracellular medium.
Fig. 2

Test of the assumptions of the formalism using a ball-and-stick model. (a) Top: scheme of the model. Excitatory (blue) and inhibitory (red) random synaptic conductances (model taken from Destexhe et al. (2001)) were inserted in the distal compartment of a ball-and-stick model with 16 compartments (see Section 2). Middle: LFP calculated from the total membrane currents, at 50 \(\upmu\)m from the neuron. Middle: PSD of the LFP, which showed a marked filtering due to the neuronal morphology, as described previously (Pettersen and Einevoll 2008). Bottom: PSDs obtained from the LFP of each individual source. (b) Same simulation as in (a), but the random synaptic inputs were distributed in all dendritic compartments and were matched to the the conductance state observed in vivo. Middle: the filtering due to morphology was negligible in this case. Bottom: individual PSDs are very similar, suggesting that a mean-field approximation is justified in this case. The dotted curves represent the optimal fit to the PSD in (b), using a Lorentzian function 1 / (1 + ωnτn) with n = 1.8 and rescaled to the maximum of each PSD

To test the validity of a mean-field approximation, we have plotted the PSD of the LFP generated by each individual source in the ball-and-stick neuron (Fig. 2(a, b), bottom panels). In the case the synaptic current is located at a single dendritic location, the PSDs of individual sources are all different (Fig. 2(a), bottom), because of the filtering due to the morphology. However, in the case of distributed current sources, the PSDs of different current sources are very similar (Fig. 2(b), bottom). This similarity also extended to LFPs calculated from different locations in extracellular space, which produced LFPs of different amplitudes, but with almost superimposable PSDs (not shown). These simulations suggest that, for in in vivo–like states, a mean-field approximation is reasonable.

3.3 Membrane impedance for a spherical source in the linear regime

In the previous sections, we assumed that the LFP generated in desynchronized network states has equivalent properties to that produced by a “mean current source”. We now calculate the membrane impedance for an individual current source of spherical symmetry, embedded in a medium which is homogeneous/isotropic and continuous, and with electric parameters σf, εf.2 The cell is represented by an RC circuit where the membrane potential is expressed as the difference Vm = Vi − Ve ≈ − 70 mV (Fig. 1(b)). At the resting membrane potential, there is a net negative charge Q− inside, which is perfectly balanced with a net positive charge Q +  on the external surface of the membrane, such that Q− + Q +  = 0. In this situation, the electric field E produced by the cell in the medium is null, which implies that \(V_e = V_f^{\infty} = 0\), where \(V_f^{\infty}\) is the reference potential at infinite distance from the source.

Suppose that a small excess of positive charge is injected inside the cell. Due to this excess of charge, an electric field E will instantaneously appear in extracellular space. An electric current will also appear to restore the equilibrium (and therefore will give rise to a variation of the electric field). The current produced depends on the physical and biological characteristics of the membrane, which will determine the time evolution of the electric field. The current is given by:
$$\begin{array}{lll} I_r &=& \sum\limits_{i=1}^{N} g_i~(t,V_{m}) (V_{m}(t)-E_i)\\ I_c &=& C_{m}\frac{d V_{m}}{dt}\\ I_{m} &=& I_r +I_c\\ \end{array} $$
(5)
where Ir and Ic are the ionic and capacitive currents, respectively. The index i represents the different membrane conductances gi, with their reversal potential Ei, and Cm is the membrane capacitance.

In the following, we will assume that the excess of charge remains small and varies around a stationary mean value, so that we can can neglect the time variations of conductances and of the VmVm), with \({\partial g_j^i}/{\partial V_{m}}\approx 0\). In this case, we can consider that the conductances are only function of the mean value of the Vm. This is equivalent to assume that, in the subthreshold regime, the conductances are not dependent on the potential, and that the reversal potentials Ei are constant. This is valid if the impact of ionic concentration changes on Ei are negligible compared to the voltage variations.

Under these approximations, we can write:
$$\begin{array}{lll} \Delta I_r &=& \sum\limits_{i=1}^{N}g_i~({\left\langle{{V_{m}}}\right\rangle}|_t) \Delta V_{m} \\ \Delta I_c &=& C_{m}\frac{d \Delta V_{m}}{dt}\\ \end{array} $$
(6)
$$ \Delta I_{m} = \Delta I_r +\Delta I_c $$
(7)
Here, we have a linear system of equations 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:
$$\begin{array}{lll} \Delta I_r(f) &=& G_{m} \Delta V_{m}(f)\\ \Delta I_c(f) &=& i \omega C_{m} \Delta V_{m}(f)\\ \Delta I_{m}(f) &=& \Delta I_r(f) +\Delta I_c(f) \end{array} $$
(8)
where
$$ G_{m}=\sum\limits_{i=1}^{N}g_{i} . $$
The impedance of the membrane is given by:
$$ Z_{m}(f)=\frac{\Delta V_{m}(f)}{\Delta I(f)} = =\frac{R_{m}}{1+i\omega \tau_{m}} \label{Zm} $$
(9)
where \(R_{m} =\frac{1}{G_{m}}\).

We next define the theoretical transfer function, which provides a relation (in this case, in the frequency domain) between LFP and Vm, as \(F_{T}({\bf r},f)=\frac{V_{m}}{V_{lfp}}({\bf r},f)\). Having defined above the impedance of the membrane we now require to define the impedance of the medium to fully characterize the transfer function. Consequently, in subsequent sections, we define the transfer function in general form and we then consider particular cases due to different recordings montages (bipolar and monopolar) and various types of media.

3.4 Computing the transfer function of each current source

In this section, we calculate separately the two transfer functions:
$$ F_{T}^{(1)}(f)=\frac{V_{m}(f)}{V_e(f)} $$
and
$$ F_{T}^{(2)}(d,f)=\frac{V_e(f)}{V_{\rm LFP}(d,f)} ~ . $$
\(F_{T}^{(1)}(f)\)3 can be calculated from the equivalent circuit shown in Fig. 1(c). We have the following relations:
$$Z_{\rm med}(R,f) = Z_a(d,f) + Z_b(d,f) \\ $$
(10)
$$V_{m}(f) = V_i(f) - V_e(f) \\ $$
(11)
$$\frac{V_i(f)}{Z_{m}(f) + Z_{\rm med}(R,f)} = \frac{V_e(f)}{Z_{\rm med}(R,f)} $$
(12)
where R is the radius of the current source and d is the distance relative to the center of the source. It follows that
$$ F_{T}^{(1)}(f)=\frac{V_{m}(f)}{V_e(f)} =\frac{Z_{m}(f)}{Z_{\rm med}(R,f)} $$
(13)
\(F_{T}^{(2)}(d,f)\) can also be calculated based on the equivalent circuit of Fig. 1(c):
$$ F_{T}^{(2)}(d,f)=\frac{V_e(f)}{V_{\rm LFP}(d,f)} =\frac{Z_a(f)+Z_b(d,f)}{Z_b(d,f)} $$
(14)
We note that if Za and Zb have the same frequency dependence, for example fn, then \(F_{T}^{(2)}\) is independent of frequency when the medium is macroscopically homogeneous. For example, if both media have a Warburg impedance (\(Z \sim 1/\sqrt{f})\) or capacitive impedance (Z ~1/f), the function \(F_{T}^{(2)}\) has the same value as for the case of a resistive medium (frequency independent impedance). For a spherical source and for an extracellular position at a distance d from the center of the source, we have:
$$ F_{T}^{(2)}({\bf r},f)= \frac{V_e(f)}{V_{\rm LFP}({\bf r},f)}=\frac{d}{R} $$
(15)
where R is the radius of the current source. In this relation, the resistance of a spherical shell of radius r in an isotropic and homogeneous medium of infinite dimension equals \(\frac{1}{4\pi\sigma r}\), which corresponds to the sum Za + Zb when r = R and Zb when r = d.
In the case of a heterogeneous isotropic medium, we have (Bédard and Destexhe 2009):
$$ Z(r,f)=\frac{1}{4\pi\sigma_z(R)} \int_{r}^{\infty}dr'\frac{1}{r'^2}~ \frac{\sigma_{f}(R)+i\omega~\epsilon_{f}(R)} {\sigma_{f}(r') + i\omega~\epsilon_{f}(r')} \label{impedance} $$
(16)

for a spherical and isopotential source. σz represents the complex conductivity. In the absence of spherical symmetry and with non-isopotential sources, it is necessary, in general, to solve differential or integral equations derived from Maxwell equations in the quasi-static regime (neglecting electromagnetic induction; see Chari and Salon 1999).

It is important to note that the expressions above for the transfer function are independent of the particular frequency spectrum of the current sources.

3.5 Frequency dependence of differential (bipolar) recordings

If the power spectral density (PSD) of the LFP signal varies as 1/fγ where γ ≥ 1 (for monopolar LFP recordings), then the energy associated to the signal would necessarily be infinite, which is of course physically impossible. In fact, according to a model developed previously (see Eqs. (53) and (54) in Bédard and Destexhe 2009), a more accurate relationship between the extracellular medium’s properties and the mean Vf is given by the following expression
$$ V({\bf r},f)= \frac{\kappa_i({\bf r})}{f^{\gamma/2}+ a_i} ~ . $$
(17)
where ai is negligible for frequencies larger than 1 Hz and smaller than about 500 Hz (because LFPs are usually considered up to 500 Hz). Note that the PSD is proportional to the square of V(f) and will thus scale as 1/fγ in this case. The PSD is also independent of the position (homogeneous medium).
The values of constants κi and ai represent respectively the proportionality constant for each electrode i, which depends on the intensity of the field for large frequencies, and the natural limit of the value of the voltage for very low frequencies (which limits the energy of the system). In general, these constants depend on electrode position, and therefore when one takes the difference between two electrodes, we have:
$$\begin{array}{lll} V_{\rm diff}(f)&=&V_{\rm LFP}^{(1)}(f)-V_{\rm LFP}^{(2)}(f)\\ &=& \frac{\kappa_1}{f^{\gamma/2}+a_1}-\frac{\kappa_2}{f^{\gamma/2}+a_2}. \end{array} $$
(18)
If the signal intensities of the two electrodes are comparable for large frequencies, we have necessarily κ1 ≈ κ2, such that the differential or bipolar signal (difference between two nearby extracellular electrodes (very correlated signals), will have the following form, on average:
$$ V_{\rm diff}(f)\approx \kappa_1\cdot\frac{a_2-a_1}{(f^{\gamma/2}+a_2)(f^{\gamma/2}+a_1)} \approx \frac{\hat{\kappa}}{f^{\gamma}}. \label{diffeq} $$
(19)
where \(\hat{\kappa} = \kappa_1(a_2-a_1)\) and for f > 1 Hz and f < 500 Hz. Thus, if monopolar LFPs have a PSD which varies as 1/fγ, one can have a PSD in 1/f2γ in bipolar recordings.

4 Numerical simulations of transfer functions

In this section, we present simple numerical simulations to illustrate how the transfer function is influenced by the frequency dependence of the cellular membrane, that of the medium, and of the recording configuration.

4.1 Resistive membrane with homogeneous/isotropic resistive medium

As a first and simplest case, suppose we have a resistive membrane (the membrane capacitance is neglected), embedded in a homogeneous resistive medium. In this case, the resistive medium is described by Laplace equation, and we have:
$$ F_{T}(f) = F_{T}^{(1)}(f)\cdot F_{T}^{(2)}(f)=\frac{R_{m}}{R_{\rm med}}\cdot\frac{R}{d}, $$
(20)
where R is the radius of the source, Rm is the membrane resistance, Rmed is the resistance of the medium, and d is the distance from the LFP measurement site to the center of the source.

This case, however, is not very realistic because the membrane capacitance is neglected. In the following sections, we consider more elaborate membranes and different extracellular media.

4.2 Capacitive effects of membranes in homogeneous/isotropic resistive media

We now consider a membrane with capacitive effects described by a simple RC circuit, together with a resistive medium. In this case, we have:
$$ F_{T}(f) = F_{T}^{(1)}(f)\cdot F_{T}^{(2)}(f)=\frac{Z_{m}(f)}{R_{\rm med}}\cdot\frac{R}{d}, $$
(21)
where the parameters are as described above, with Zm(f) the membrane impedance. Thus, according to Eq. (9), we have the following transfer function:
$$ F_{T}^s(f) = \frac{R_{m}}{R_{\rm med}}\cdot\frac{1}{1+i\omega \tau_{m}}\cdot\frac{R}{d}. \label{RC_resisitveFT} $$
(22)
where τm is the membrane time constant. This transfer function is depicted in Fig. 3(a).
Fig. 3

Amplitude and phase of the transfer function FT as a function of frequency for different models and for mono-polar electrode montage. In all cases the transfer function was estimated for a distance of \(30~\upmu {\rm m}\) from a spherical source of 10 \(\upmu {\rm m}\) radius. (a) Standard RC membrane model with \(C_{m}=10^{-2}~{\rm F/m}^2\) and various configurations for membrane time constant τm. The extracellular medium conductivity is 0.3 S/m. (b) Non-ideal membrane model (see Bédard and Destexhe 2008) with same parameters as in (a), and with a Maxwell-Wagner time τMW = 5 ms (see Appendix A). The extracellular medium is also resistive in this case. Surprisingly, we observe a resonance in the phase of the transfer function as indicated by (⋆). (c) Standard RC membrane model together with a medium described by a Warburg impedance \(\frac{e^{i\phi}}{4\pi\sigma\sqrt{f}}\) where ϕ is frequency independent. We have taken ϕ = 0 to illustrate the differences between this type of impedance compared to a resistive medium. ϕ will be different than zero in general, with no change to the modulus. Note that there is a peak in amplitude of the absolute value of the transfer function that increases and shifts to lower frequencies as τm increases; ⋆ indicates a resonance

The transfer function can also be calculated for a “non ideal” membrane, with a more realistic RC circuit model where the capacitance is non-ideal and does not charge instantaneously (see details in Bédard and Destexhe (2008); see also Appendix A). Considering such a non-ideal membrane with a resistive medium, we have:
$$ F_{T}^{N}(f) = \frac{R_{m}}{R_{\rm med}}\cdot\frac{1}{1+i\frac{\omega \tau_{m}}{1+i\omega \tau_{\rm MW}}}\cdot\frac{R}{d} \label{nst} $$
(23)
where τMW is the Maxwell-Wagner time of a non-ideal capacitance. Note that when τMW = 0, we recover the case above for an ideal membrane. This transfer function is represented in Fig. 3(a). The transfer function is in general monotonic (and scales close to 1/f2).

Interestingly, there is a phase resonance for non-ideal membranes (see (⋆) in Fig. 3(b)). The physical origin of this phase resonance in the non-ideal cable is that the membrane is quasi-resistive at low (≃ 0) and high frequencies (100 Hz) (see Eq. (33)), so that the absolute value of the phase must necessarily pass through a maximum.4

4.3 RC membrane in homogeneous/isotropic non-resistive medium

We now focus on non-resistive media by providing a transfer function for which the functional form should be observable from extracellular measurements. We consider the simplest case scenario of only linear subthreshold regime where the membrane is described by a simple RC circuit embedded in a medium with impedance Z(f). In this case, the transfer function for a mono-polar extracellular recording is given by:
$$ F_{T}(f) = R_{m} \cdot \frac{1}{ Z(f) \ (1+i\omega \tau_{m})}\cdot\frac{R}{d}, \label{GeneralMonoFT} $$
(24)
If the impedance of the medium is \(\frac{\kappa}{f^{\gamma/2}-a_i}\) and γ ≥ 1, where κ is a complex constant, the monopolar and bipolar transfer functions are respectively:
$$ \begin{array}{ccc} F_{T}^{\rm mono}(f) &\approx & \frac{R_{m}}{\kappa}\cdot\frac{(f)^{\gamma/2}}{1+i\omega \tau_{m}}\cdot\frac{R}{d}.\\ F_{T}^{\rm diff}(f) &\approx & \frac{R_{m}}{\hat{\kappa}}\cdot\frac{f^{\gamma}}{1+i\omega \tau_{m}}\cdot\frac{R}{d}. \label{WarburgMonopolarFT} \end{array} $$
(25)

The differences between monopolar and bipolar transfer functions are explained in Section 3.5. In Fig. 3(c), we show few examples of such a case with different values of the membrane time constant τm. We observe that the modulus of the transfer function can present a maximum which depends on τm, and therefore of the level of activity or the “conductance state” of the membrane.

5 Comparison with experimental results

In this section, we test the resistive or non-resistive nature of the extracellular medium by evaluating the transfer function from experimental data and compare with theoretical estimates. However, we note that in our experiments we use bipolar LFPs recorded in rat barrel cortex, simultaneously with intracellular recordings in the same cortical area. Since the LFP recordings are bipolar, we use the following form (see Section 4.3):
$$ F_{T}^{\rm (dif\/f)}(f) = \frac{R_{m} ~f^{\gamma}}{\hat{\kappa}\ (1+i\omega \tau_{m})}\cdot\frac{R}{d}, \label{GeneralDiffFT} $$
(26)
with \(Z(f) = \frac{\kappa}{f^{\gamma/2}-a_i}\) and γ ≥ 1. Here, d is the distance between the recording site and the source (approximately 1 mm in these experiments).
In the particular case of a Warburg impedance (γ = 1), we have:
$$ F_{T}^{\rm (dif\/f)}(f) = \frac{R_{m}~f}{\hat{\kappa}_{w}(1+i\omega \tau_{m})}\cdot\frac{R}{d}, \label{WarburgDiffFT} $$
(27)
where \(\hat{\kappa}_{w}\) is a complex constant because a Warburg impedance is such that \(Z(f) = \frac{a+ib}{\sqrt{f}}\). For a quasi-resistive medium \(Z(f) = \frac{\kappa}{f^{\gamma}-a_i}\) with γ very close to zero, we have:
$$ F_{T}^{\rm (dif\/f)}(f) = \frac{R_{m}}{\hat{\kappa_{r}}\ (1+i\omega \tau_{m})}\cdot\frac{R}{d}, \label{ResistiveDiffFT} $$
(28)
where \(\hat{\kappa}_{r}\) is a real constant.
Finally, in the case of a purely capacitive medium, γ = 2 and we have:
$$ F_{T}^{\rm (dif\/f)}(f) = \frac{R_{m}~f^2}{\hat{\kappa}_c\ (1+i\omega \tau_{m})}\cdot\frac{R}{d}, \label{CapacitiveDiffFT} $$
(29)
where \(\hat{\kappa}_{r}\) is a purely imaginary constant in this case.
We now compare Eqs. (27)–(29) with the experimental measurements. We have analyzed four neurons in which simultaneous Vm and (bipolar) VLFP were obtained from rat barrel cortex in vivo. Because the theoretical estimates are for linear regime activity, our analysis must avoid any possible interference with spikes, and focus solely on long periods of subthreshold activity as marked by the grey shaded boxes superimposed on the Vm and LFP traces (see Fig. 4). The bottom panels of Fig. 4 show the PSD of the Vm and of VLFP, which display similar frequency-scaling exponents. Note that the exponent values of Vm activity lie within the range identified for other anesthetic conditions, for which the Vm exponent varies between − 2 and − 3 (El Boustani et al. 2009; Rudolph et al. 2005).
Fig. 4

Power spectra of simultaneous intracellular and LFP recordings in desynchronized states in vivo. The top panels depict the time series of simultaneously recorded bipolar LFP and Vm from rat barrel cortex in light anesthesia with low-amplitude desynchronized EEG (all recordings at zero current). The shaded grey box indicate the time period (T ≈ 38.69 s) of subthreshold activity selected for analysis. The bottom panels are the calculated Power spectra (PSD) of both Vm (top plot in dark grey) and LFP (bottom plot in light grey), which show similar scaling. Superimposed, in black, are the moving average PSD with a window of ~7.7 sec (see Section 2). This procedure results in a PSD with reduced variance but also reduced frequency resolution. The total number of points analyzed was \(N=386\mathord{,}900\) (between 218 and 219). The similar scaling between Vm and LFP is highlighted by the overlaid white lines that have exactly slope = − 2. In particular, the Vm scales with an exponent comprised between − 2 and − 2.4, while the LFP exponents range between − 1.9 and − 2.6.

To compute the transfer function from these data sets, we evaluate the ratio between the absolute value of the Fourier transform of both Vm and LFP as shown in Fig. 5(a) (light grey curves), which corresponds to the data of Fig. 4. As suggested by this similar scaling (bottom panel of Fig. 4), the transfer function of the data \(F_{T}^{\rm (dif\/f)}\) has a mean value that is approximately constant (slope zero) for a large frequency range (about 10–500 Hz) (Fig. 5(a, b)).
Fig. 5

Transfer function \(F_{T}^{\rm (dif\/f)}\) computed from experimental data. The top panel corresponds to the cell shown in Fig. 4 and the bottom panel to a different cell. The experimentally calculated \(F_{T}^{\rm (dif\/f)}\) (shown in light grey) shows an average slope of zero for frequencies between about 3 Hz and 500 Hz, and is compared to the best fits using a Warburg-type medium (solid black line), a resistive medium (black dashed line) and purely capacitive medium (red dashed line). Two different methods were used to calculate the best nonlinear least square fit, a polynomial averaging algorithm (main plots, third-order polynomial average shown as dashed blue curves; see Appendix B for details) and a moving average method which reduces the variance of the PSD (insets; see Section 2). In both cases, the fit was constrained to frequencies between 3 and 500 Hz. The parameters of the respective fits are given in Table 1.

Table 1

Parameters for the fitting of the transfer function to experimental data for different types of extracellular impedance

Cell

Impedance type

ε

τm (ms)

α

1

Warburg

3.4×102

17.5

1.43

Resistive

\(\phantom{0.}5\times 10^3\)

5

194.94

Capacitive

\(\phantom{0.}1\times 10^8\)

15

0.001

1*

Warburg

3.2×105

14.8

1.22

Resistive

1.3×106

5

111.74

Capacitive

4.3×107

15

0.010

2

Warburg

3.4×102

24.5

1.3

Resistive

\(\phantom{0.}5\times 10^3\)

5

186.10

Capacitive

7.8×107

50

0.004

2*

Warburg

1.4×106

24.4

1.28

Resistive

1.7×106

5

63.84

Capacitive

9.3×106

50

0.002

\(\epsilon = ||y - \hat{y}(\tau_{m}, \alpha) ||_2\): squared 2-norm of the fit residual, where y denotes the data and \(\hat{y}(\tau_{m}, \alpha)\) represents the various models (Warburg, Resistive and Capacitive). τm: membrane time constant and \(\alpha = \frac{R_{m} R}{\hat{k}d}\) (refer back to equations in the main text), which are obtained by the fitting procedure. Cells 1 and 2 correspond to (a) and (b) in Fig. 5; Cells 1* and 2* refer to the values obtained in the same cells using the moving average method (Fig. 5, insets)

We performed a constrained nonlinear least square fit for the three transfer functions (Eqs. (27), (28) and (29)) to the calculated data transfer function \(F_{T}^{\rm (dif\/f)}\) for the frequency range between 3 and 500 Hz. The membrane time constant was constrained to physiological range, τm ∈ [5 ms, 50 ms], whereas the lumped parameter, \(\alpha=\frac{R_{m} R}{\hat{k} d}\), was allowed to vary for a large set, α ∈ [0, 103]. Note that we chose not to fit all the parameters (Rm, R, \(\hat{k}\) and d) as some of these parameters are related (e.g. Rm and τm) and there is an indeterminacy about these parameters because they are lumped. In addition, some parameters will vary from experiment to experiment. For instance, electrode coefficients, size of the measured cell, diffusion constants (embedded in the parameter \(\hat{k}\), as indicated by Eqs. (53) and (54) in Bédard and Destexhe 2009) will vary across different experiments. To fit all these parameters, a different experimental protocol would be required to gather the necessary data to disambiguate them. Hence, we decided to lump them into the single parameter α and also for the fitting purpose we consider α and τm to be independent.

To ensure soundness in the parameter fitting, we employed two different averaging techniques to the transfer function \(F_{T}^{\rm diff}\) (see Section 2). Polynomial averaging techniques (see Appendix B) are known to produce robust results when the variance of the signal is very high Press et al. (2008). This method is applied here in the frequency domain, and virtually suppresses all of the variance of the transfer function (Fig. 5(a, b), main plots; see the dashed magenta curves). In addition, we also used a moving average window procedure (see Section 2), which markedly reduces the variance (Fig. 5, insets; see dark grey shaded curves superimposed on the original PSD in light grey).

The theoretical expressions for the transfer function were fit to the transfer functions reconstructed from the experiments. These fits were performed for a Warburg-type medium (Fig. 5, solid lines), for a resistive medium (dashed lines) and for purely capacitive medium (dashed red lines). Constrained to the frequency range of 3–500 Hz, the fit was always markedly better for Warburg type impedances, for both cells shown in Fig. 5 and for both methods, as shown by the values of the residuals (see Table 1). Note that for a resistive-type medium, as well as for a purely capacitive medium, the parameter estimation (in particular for τm) always reached the boundaries of the constrained values, which is indicative that the obtained minimum in parameter space was far from optimal (even if the time constants may coincidentally reach realistic values). To confirm this, we allowed an unconstrained parameter fit and we observed that although the resistive-type would seemingly improve the fit (i.e., the initial plateau of the resistive-type curve would shift towards the 10–500 Hz range), it would however provide unrealistic parameter ranges. For the purely capacitive case, it was always impossible to achieve a good fit (i.e., slope zero). This is to be expected as the numerator of Eq. (29) grows quadratically and faster than the denominator, hence it behaves like the identity function, y = f, as frequency tends to infinity. Also note that the polynomial averaging algorithm of Appendix B always gave better results (see values of the residuals in Table 1), although performed on a dataset of higher variance (compare the grey lines in the main plots and insets of Fig. 5).

Note that the large difference in the residuals between the polynomial and moving average fits, in particular for the Wargburg Impedance, can be explained by the fact that the moving average does not fully eliminate the variance of the signal, while all the variance is virtually removed by the polynomial method (see magenta curves in Fig. 5(a, b)). The error increases even further for larger frequencies as the slope starts to change, possibly due to other phenomena, which the model does not capture. Also note that polynomial and moving average methods were compared in a previous study (see Chapter 14 and Fig. 14.8.1 in Press et al. 2008), which should be consulted for details. We stress that for the Warburg type medium that values obtained for τm and α are consistent with the data and theoretical predictions. In particular, the theory with Warburg impedance predicts that the values of α should be negligible for frequencies greater than 1 Hz (see Eqs. (53) and (54) in Appendix of Bédard and Destexhe 2009).

Similar results were also obtained for two other cells of the same database (not shown). The same results were also found by performing a similar analysis on another data set consisting of simultaneous intracellular and LFP recordings in awake cats (not shown; data from Steriade et al. (2001), courtesy of Igor Timofeev, Laval University, Canada).

6 Discussion

In this paper, we have examined the transfer function between intracellular and extracellular potentials. By using a mean-field approximation in Fourier frequency space, we derived a method allowing us to obtain an expression relating the LFP with the intracellular Vm activity. The main theoretical finding is that this transfer function (which does not depend on the frequency spectrum of current sources) takes very different forms according to the type of frequency dependence of the extracellular medium, and thus could be used as a means to estimate which type of frequency dependence (if any) is most consistent with experiments. Second, we have applied this formalism to intracellular recordings in desynchronized EEG states, for which the mean-field approximation should best apply. We found that, in rat barrel cortex, the extracellular medium seems frequency dependent with a Warburg type impedance.

One key assumption of the present formalism is that individual synaptic currents sources are uncorrelated. There is ample evidence that this is the case for EEG-desynchronized states, as shown by the low levels of correlation between simultaneously recorded units (Contreras and Steriade 1996; Destexhe et al. 1999; Gawne and Richmond 1993; Steriade 2001; Zohary et al. 1994), or by the low correlations between multi-site LFPs (Destexhe et al. 1999). However, there are limits to this assumption. First, dual recordings in awake mice barrel cortex showed that the subthreshold activities of neurons can display periods of significant correlation, even with desynchronized EEG Poulet and Petersen (2008). Indeed, a certain level of correlation is unavoidable from the redundant connectivity of neurons in cortex, although the activity itself can produce negative correlations which may cancel the effect of redundant connectivity (Renart et al. 2010). Second, it is evident that the activity cannot be totally decorrelated, otherwise the EEG amplitude would be close to zero (Gold et al. 2006). Nevertheless, the amplitude of the EEG during desynchronized states is considerably lower than during synchronized activity (such as slow waves), which is associated to a general decorrelation of neurons (Contreras and Steriade 1996), and the present formalism should be applicable to such desynchronized states.Further studies should consider these points to build more realistic mean-field models of desynchronized states, which would lead to more realistic transfer functions.

In a previous investigation (Bédard and Destexhe 2009), we have shown theoretically that several physical phenomena can lead to frequency dependence of the extracellular medium: ionic diffusion and membrane polarization. The former predicts an impedance of Warburg type (\(Z \sim 1 / \sqrt{f}\)), while the latter predicts a capacitive-type impedance (Z ~1/f) (Bédard et al. 2006b). These two phenomena can also explain different experimental observations: the frequency dependent conductivity observed experimentally in brain tissue (Gabriel et al. 1996) can be reproduced by a combination of these two mechanisms. Recent measurements from monkey cortex suggesting resistive medium (Logothetis et al. 2007) can be explained by the fact that the influence of diffusion was avoided in that case. This technique is based on the saturation effect (Geddes effect, which is represented by Zener diodes in Logothetis et al. 2007), which greatly diminishes the concentration gradient around the electrode, such that the ionic diffusion is more limited. A Warburg type impedance was also found to account for the 1/f power spectral structure of LFPs (see details in Bédard and Destexhe 2009).

The present results are consistent with this analysis. The transfer functions measured here for 4 cells are all consistent with the Warburg type impedance of ionic diffusion up to 500 Hz. The other type of extracellular impedances mentioned above, either purely resistive or purely capacitive, could not fit the data (see Fig. 5). While these results seem to rule-out purely resistive or capacitive media, there is still a possibility that they apply outside the 3–500 Hz frequency range. For example, polarization phenomena, which can be modeled as a capacitive effect with a low cutoff frequency (Bédard et al. 2006b), may contribute to the low frequency range (below 10 Hz). Further theoretical and experimental work is needed to investigate these aspects. In particular, experiments should be carried with controlled current sources as close as possible to the biological current sources, for example using micropipettes.

It is important to keep in mind that the present method derives from a mean-field approach in frequency space, and thus relies on the assumption that individual current sources are independent. This justifies the use of desynchronized EEG states, in which synaptic current sources are expected to have very low correlation. The simulations presented in Fig. 2 (bottom panels) show that the LFP predicted by a compartmental dendritic model is virtually identical to that obtained by individual compartments, which suggests that considering a set of independent sources is not a bad approximation. However, if strong correlations occur, such as during synchronized population activities, the current sources may no longer be considered as independent, and another formalism should be used.

A recent study on modeling extracellular action potentials (Pettersen and Einevoll 2008) showed that the cable structure of neurons can also cause low-pass filtering, because the return current is itself filtered by the membrane capacitance. This is a clear example where the correlation between current sources cannot be neglected (see also Yaron-Yakoubovitch et al. 2008). However, we showed here that such a contribution is negligible for in vivo conditions (Fig. 2(a, b)), presumably because the axial currents are very small compared to the intense synaptic currents. So far, the only plausible physical cause to explain the observed 1/f filtering under in vivo conditions is ionic diffusion (for frequencies up to 500 Hz). It is possible that for states of reduced synaptic activity, the filtering due to morphology plays a role, although this still needs to be demonstrated experimentally Further studies should investigate these aspects by constraining these different theories by appropriate experiments.

One major criticism to the previous measurement techniques (Ranck 1963; Gabriel et al. 1996; Logothetis et al. 2007) is that they use current intensities of one or several order of magnitude larger than biological sources, and these currents evidently interact with the medium very differently as natural sources. The present method has the advantage of not suffering from this limitation, because it is using only passive recordings of physiological signals, with no need of injecting currents. This method should therefore be considered as complementary to direct measurements.

Finally, the expression given by Eqs. (24) and (26) could be used to directly estimate the impedance of the extracellular medium as a function of frequency (still within the mean-field approximation). We did not attempt this type of approach here, but instead considered different hypotheses concerning the impedance of the medium. The present analysis reported here for 4 cells was also confirmed by using two cells from another database of intracellular recordings in desynchronized EEG states in awake animals (courtesy of I. Timofeev, Laval University, Canada), which also indicated a Warburg type impedance (not shown). The same approach should be extended to a much larger sample of cells and brain states, to provide a full estimate of the impedance spectrum of the medium. The present results therefore must be considered as preliminary and must be confirmed by using further analyses of simultaneously recorded LFPs and intracellular recordings during desynchronized EEG states in vivo.

Footnotes

  1. 1.

    Note that for desynchronized activity, we expect that such a spatial average will be of small amplitude, as indeed typically found for the “desynchronized EEG” condition investigated here.

  2. 2.

    Note that we keep the electric parameters frequency dependent, to keep the expressions as general as possible. In addition, the theory can easily be generalized to multipoles, as any multipole configuration can be decomposed in a sum of monopoles, and to multiple sources using the linear superposition principle.

  3. 3.

    Note that the notation \(F_{T}^{(n)}\) stands for different functions and not for nth order derivative.

  4. 4.

    Every positive continuous function defined on a compact domain has necessarily a maximum inside that domain.

Notes

Acknowledgements

Research supported by the Centre National de la Recherche Scientifique (CNRS, France), Agence Nationale de la Recherche (ANR, France) and the Future and Emerging Technologies program (FET, European Union; FACETS project). Additional information is available at http://cns.iaf.cnrs-gif.fr.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Claude Bédard
    • 1
  • Serafim Rodrigues
    • 1
  • Noah Roy
    • 2
  • Diego Contreras
    • 2
  • Alain Destexhe
    • 1
  1. 1.Integrative and Computational Neuroscience Unit (UNIC), UPR2191CNRSGif-sur-YvetteFrance
  2. 2.Department of NeuroscienceUniversity of PennsylvaniaPhiladelphiaUSA

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