Macroscopic Models of Local Field Potentials and the Apparent 1/f Noise in Brain Activity

Transcription

1 Biohysical Jornal Volme 96 Aril Macroscoic Models of Local Field Potentials and the Aarent /f Noise in Brain Activity Clade Bédard and Alain Destexhe* Integrative and Comtational Neroscience Unit (UNIC ), Centre National de la Recherche Scientifiqe, Gif-sr-Yvette, France ABSTRACT The ower sectrm of local field otentials (LFPs) has been reorted to scale as the inverse of the freqency, bt the origin of this /f noise is at resent nclear. Macroscoic measrements in cortical tisse demonstrated that electric condctivity (as well as ermittivity) is freqency-deendent, while other measrements failed to evidence any deendence on freqency. In this article, we roose a model of the genesis of LFPs that acconts for the above data and contradictions. Starting from first rinciles (Maxwell eqations), we introdce a macroscoic formalism in which macroscoic measrements are natrally incororated, and also examine different hysical cases for the freqency deendence. We sggest that ionic diffsion rimes over electric field effects, and is resonsible for the freqency deendence. This exlains the contradictory observations, and also rerodces the /f ower sectral strctre of LFPs, as well as more comlex freqency scaling. Finally, we sggest a measrement method to reveal the freqency deendence of crrent roagation in biological tisse, and which cold be sed to directly test the redictions of this formalism. INTRODUCTION Macroscoic measrements of brain activity, sch as the electroencehalogram (EEG), magnetoencehalogram or local field otentials (LFPs), dislay ~/f freqency scaling in their ower sectra ( 4). The origin of sch /f noise is at resent nclear. The /f sectra can reslt from selforganized critical henomena (5), sggesting that neronal activity may be working according to sch states (6). Alternatively, the /f scaling may be de to filtering roerties of the crrents throgh extracelllar media (2). The latter hyothesis, however, was resting on indirect evidence, and still needs to be examined theoretically, which is one of the motivations of this article. A continm model (7) of LFPs incororated the inhomogeneities of the extracelllar medim into continos satial variations of condctivity (s) and ermittivity (3) arameters. This model rerodced a form of low-ass freqency filtering in some conditions, while considering the extracelllar medim as locally netral with s and 3 arameters indeendent of freqency. This model was not entirely correct, however, becase macroscoic measrements in cortex revealed a freqency deendence of electrical arameters (8). We will show here that it is ossible to kee the same model strctre by inclding lasible cases for the freqency deendence. In a olarization model (9) of LFPs, the variations of condctivity and ermittivity were considered by exlicitly taking into accont the resence of varios celllar rocesses in the extracelllar sace arond the crrent sorce. In articlar, it was fond that the henomenon of srface olarization was fndamental to exlain the freqency deendence Sbmitted Agst 28, 2008, and acceted for blication December 3, *Corresondence: destexhe@nic.cnrs-gif.fr of LFPs. The continm model (7) incororated this effect henomenologically throgh continos variations of s and 3. In the olarization model, the extracelllar medim is reactive in the sense that it reacts to the electric field by olarization effects. It is also locally nonnetral, which enables one to take into accont the noninstantaneos character of olarization, which is at the origin of freqency deendence according to this model (9). In this article, we roose a diffsion-olarization model that synthesizes these revios aroaches and which takes into accont both microscoic and macroscoic measrements. This model incldes ionic diffsion, which we will show has a determinant inflence on freqency filtering roerties. The model also incldes electric olarization, which also inflences the freqency-deendent electric roerties of the tisse. We show that taking into accont ionic diffsion and electric olarization allows s to qantitatively accont for the macroscoic measrements of electric condctivity in cortical tisse according to the exeriments of Gabriel et al. (8). However, recent measrements of Logothetis et al. (0) evidenced that the freqency deendence of cortical tisse was negligible, therefore in contradiction with the measrements of Gabriel et al. (8). We show here that the diffsionolarization model can be consistent with both tyes of exeriments, and ths may reconcile this contradiction. We will also examine whether this model can also exlain the /f freqency scaling observed in LFP or EEG ower sectra. Finally, we consider ossible ways for exerimental test of the redictions of this model. The final goal of this aroach is to obtain a model of local field otentials, which is consistent with both macroscoic measrements of condctivity and ermittivity, and the microscoic featres of the strctre of the extracelllar sace arond the crrent sorces. Editor: Arthr Sherman. Ó 2009 by the Biohysical Society /09/04/2589/5 $2.00 doi: 0.06/j.bj

2 2590 Bédard and Destexhe MATERIALS AND METHODS Nmerical Simlation of Macroscoic Measrements (see Nmerical Simlation of Macroscoic Measrements, below) describes the imedance of the extracelllar medim based on Zðr ; Þ ¼ 4 dr 0 r r 0 2 s M ðr 0 ; Þ þi3 M ðr 0 ; Þ : () This eqation gives the -freqency comonent of the imedance at oint r in extracelllar sace, in sherical symmetry (see (7) and Eq. for details). To evalate this eqation, we se MATLAB (The MathWorks, Natick, MA), which comtes the Riemann sm, Zðr ; Þ ¼ X N 4 r Dr0 r 0 2 s M ðr 0 ; Þ þi3 M ðr 0 ; Þ ; (2) where Dr 0 is the integration ste ( mm) and N is determined for a slice of mm. We also se the arametric model of Gabriel et al. (8) to simlate the freqency deendence of electrical arameters s and 3 of the extracelllar flid from gray matter (at a temeratre of 37 C). This model is valid for freqencies inclded in the range of 0 Hz to Hz (). According to this model, the absolte comlex and macroscoic ermittivity and condctivity (measred between 0 and 0 0 Hz) in cortical gray matter is given by the Cole-Cole arametric model (2), 3 s ¼ i ¼ 3 X n ¼ 4 D3 n N3 o þ 3 o þðit n Þ i s an ; (3) n ¼ where the sm rns over for olarization modes n, 3 o ¼ F/m is vacm ermittivity, 3 N ¼ 4.0 is the ermittivity relative to f ¼ 0 0 Hz, s ¼ 0.02 S/m is the static condctivity at f ¼ 0 Hz according to the chosen arametric model. The arameters nder the sm of Eq. 3 are given in Table. RESULTS We start by otlining a macroscoic model with freqencydeendent electrical arameters (Macroscoic Model of Local Field Potentials), and we discss the main hysical cases for this freqency deendence (Physical Cases for Freqency-Deendent Electrical Parameters). We then constrain the model to macroscoic measrements of electrical arameters, and rovide nmerical simlations to test the model and rerodce the exerimental observations (Nmerical Simlation of Macroscoic Measrements). Finally, we roose a ossible way to test the model exerimentally (Measrement of Freqency Deendence). Macroscoic model of local field otentials In this section, we derive the eqations governing the time evoltion of the extracelllar otential. We follow a formalism TABLE Parameter vales for the arametric model of Gabriel et al. (8) (see Eq. 3) No. D3 n t n (s) a n similar to the one develoed reviosly (7), excet that we reformlate the model macroscoically, to allow the electrical arameters (the condctivity s and ermittivity 3) to deend on freqency, as demonstrated by macroscoic measrements (8,,3). The hysical cases of this macroscoic freqency deendence will be examined in Physical Cases for Freqency-Deendent Electrical Parameters. General formalism We begin by deriving a general eqation for the electrical otential when the electrical arameters are freqencydeendent. We start from Maxwell eqations, taking the first and the divergence of the forth Maxwell eqation in a medim with constant magnetic ermeability, giving V, ~D ¼ r free V,~j þ vrfree vt ¼ 0; where ~D, ~j and r free are, resectively, the electric dislacement, crrent density, and charge density in the medim srronding the sorces. Moreover, in a linear medim the eqations linking the electric field ~E with electric dislacement ~D, and with crrent density~j, gives and ~Dð~x; tþ ¼ ~jð~x; tþ ¼ N N (4) 3ð~x; tþ~eð~x; t tþdt (5) sð~x; tþ~eð~x; t tþdt: (6) The Forier transforms of these eqations are resectively ~D ¼ 3 ~E and ~j ¼ s ~E, where we allow s and 3 to deend on freqency. Given the limited recision of measrements, we can consider V ~Ez0 for freqencies <000 Hz. Ths, we can assme that ~E ¼ VV, sch that the comlex Forier transform of the exressions in Eq. 4 can be written as Conseqently, we have V, ð3 ð~xþvv Þ¼ r free V, ðs ð~xþvv Þ¼ir free : V, ððs þ i3 ÞVV Þ¼Vðs þ i3 Þ, VV þðs þ i3 ÞV 2 V ¼ 0: Comared to revios derivations (see Eq. 49 in (7)), this eqation is a more general form in which the electrical arameters can be deendent on freqency. Macroscoic model In rincile, it is sfficient to solve Eq. 7 in the extracelllar medim to obtain the freqency deendence of LFPs. However, in ractice, this eqation cannot be solved becase the strctre of the medim is too comlex to roerly define (7)

3 Macroscoic Model of LFPs 259 the limit conditions. The associated vales of electric arameters mst be secified for every oint of sace and for each freqency, which reresents a considerable difficlty. One way to solve this roblem is to consider a macroscoic or mean-field aroach. This aroach is jstified here by the fact that the vales measred exerimentally are averaged vales, which recision deends on the measrement techniqe. Becase or goal is to simlate those measred vales, we will se a macroscoic model, in which we take satial averages of Eq. 7, and make a continos aroximation for the satial variations of these average vales. This tye of aroximation can be fond in the classic theory of electromagnetism (5). To this end, we define macroscoic electric arameters, 3 M and s M,as and 3 M ð~xþ ¼h3 ð~xþi jv ¼ f ð~x; Þ s M ð~xþ ¼hs ð~xþi jv ¼ gð~x; Þ; where V is the volme over which the satial average is taken. We assme that V is ~mm 3, and is ths mch smaller than the cortical volme, so that the mean vales will be deendent of the osition in cortex. Becase the average vales of electric arameters are statistically indeendent of the mean vale of the electric field, we have ~ j total j V ð~x; tþ ¼ N þ s M ðtþh~ei jv ð~x; t tþdt N vh~ei 3 M jv ðtþ ð~x; t tþdt; vt where the first term on the right-hand side reresents the dissiative contribtion, and the second term reresents the reactive contribtion (reaction from the medim). Here, all hysical effects, sch as diffsion, resistive and caacitive henomena, are integrated into the freqency deendence of s M and 3 M. We will examine this freqency deendence more qantitatively in Physical Cases for Freqency- Deendent Electrical Parameters. The comlex Forier transform of h~j total i jv ð~x; tþ then becomes ~ j total j V ¼ s M þ i3m h ~E i jv ¼ s M z h ~E i jv ; (8) where s z M is the comlex condctivity. We can also assme sch that s M z ¼ i3 M z ; (9) V, ~j total j ¼ V, s Mz h ~E i jv V ¼ V, i3 Mz h ~E i jv ¼ 0: (0) Becase s M z ¼ (s M þ i3 M ) and ~E ¼ VhV i, the exressions above (Eqs. 0) can also be written in the form V, ðs M þ i3m ÞVhV i jv ¼ 0: () We note that starting from the continm model (7), where only satial variations were considered, and generalizing this model by inclding freqency-deendent electric arameters, gives the same mathematical form as the original model (comare with Eq. 49 in (7)). This form invariance will allow s to introdce, in Physical Cases for Freqency-Deendent Electrical Parameters, the srface olarization henomena by inclding an ad hoc freqency deendence in s M and 3 M. The hysical cases of this macroscoic freqency deendence is that the cortical medim is microscoically nonnetral (althogh the cortical tisse is macroscoically netral). Sch a local nonnetrality was already ostlated in a revios model of srface olarization (9). This sitation cannot be acconted for by Eq. 7 if s M and 3 M are freqency-indeendent (in which case r ¼ 0 when VV ¼ 0). Ths, inclding the freqency deendence of these arameters enables the model to catre a mch broader range of hysical henomena. Finally, a fndamental oint is that the freqency deendences of the electrical arameters s M and 3 M cannot take arbitrary vales, bt are related to each other by the Kramers-Kronig relations (7 9) and D3 M ðþ ¼ 2 0 s M ð 0 Þ 02 2d0 (2) s M ðþ ¼s M ð0þ 22 D3 M ð 0 Þ ; (3) 02 2d0 where rincial vale integrals are sed. These eqations are valid for any linear medim (i.e., when Eqs. 5 and 6 are linear). These relations will trn ot to be critical to relate the model to exeriments, as we will see below. Note that, contrary to freqency deendence, the satial deendences of s M and 3 M are indeendent of each other, becase these deendences are related to the satial distribtion of elements within the extracelllar medim. Simlified geometry for macroscoic arameters To obtain an exression for the extracelllar otential, we need to solve Eq., which is ossible analytically only if we consider a simlified geometry of the sorce and srronding medim. The first simlification is to consider the sorce as monoolar. The choice of a monoolar sorce does not intrinsically redce the validity of the reslts becase mltiolar configrations can be comosed from the arrangement of a finite nmber of monooles (20). In articlar, if the hysical natre of the extracelllar medim determines a freqency deendence for a monoolar sorce, it will also do so for mltiolar configrations. A second 0

4 2592 Bédard and Destexhe simlification will be to consider that the crrent sorce is sherical and that the otential is niform on its srface. This simlification will enable s to calclate exact exressions for the extracelllar otential and shold not affect the reslts on freqency deendence. A third simlification is to consider the extracelllar medim as isotroic. This assmtion is certainly valid within a macroscoic aroach, and jstified by the fact that the neroil of cerebral cortex is made of a qasirandom arrangement of celllar rocesses of very diverse size (2). This simlified geometry will allow s to determine how the hysical natre of the extracelllar medim can determine a freqency deendence of the LFPs, indeendently of other factors (sch as more realistic geometry, roagating otentials along dendrites, etc.). Ths, considering a sherical sorce embedded in an isotroic medim with freqency-deendent electrical arameters, combining with Eq., we have d 2 hv i jv þ 2 dhv i jv dr 2 r dr dðs þ i3 Þ dhv i jv þ ¼ 0: ðs þ i3 Þ dr dr (4) Integrating this eqation gives the following relation between two oints r and r 2 in the extracelllar sace, i rdhv 2 jv ðr Þ½s ðr Þþi3 ðr ÞŠ dr i ¼ r2dhv 2 jv ðr 2 Þ½s ðr 2 Þþi3 ðr 2 ÞŠ: (5) dr Assming that the extracelllar otential vanishes at large distances (hv i¼0), we find hv i jv ðr Þ¼ I 4 dr 0 r r 0 2 s ðr 0 Þþi 3 ðr 0 Þ : (6) This eqation is analogos to a similar exression derived reviosly (Eq. 25 in (7)), bt more general. The two formalisms are related by Physical cases for freqency-deendent electrical arameters In the following, we sccessively consider two different cases of extracelllar medim: first, nonreactive media, in which the crrent assively flows into the medim; and second, reactive media, in which some roerties (sch as charge distribtion) may change after crrent flow. For each medim, we will consider two tyes of hysical henomena the crrent rodced by the electric field, and the crrent rodced by ionic diffsion, as schematized in Fig. ). Nonreactive media with electric fields (Model N) Nonreactive media ( 3M ; s M s M ¼ sm and 3 M ¼ 3 M ) are eqivalent to resistive media, in which the resistance (or eqivalently, the condctivity) does not change after the flow of crrent. The simlest tye of sch configration consists of a resistive medim (sch as a homogeneos condctive flid) in which crrent sorces solely interact via their electric field. Alying Eq. 7 to this configration is eqivalent to model the extracelllar otential by Colomb s law, V ð~rþ ¼ 4s M, I r ; (8) where V ð~rþ is the extracelllar otential at a osition defined by ~r in extracelllar sace, and r is the absolte distance between ~r and the center of the crrent sorce. Here, the condctivity (s M (r) ¼ s M ) is indeendent of sace and freqency, and ths, this model is not comatible with macroscoic measrements of freqency deendence (8,,3). h~j i jv ¼ s z h~e i jv ¼ðs M þ i3m Þh ~E i jv instead of ~j ¼ s M ~E (see Eq. 4 in (7)). This difference is becase the condctivity here deends on freqency. In the following, we will se the simlified notations ~j ;~E and V instead of h~j i jv, h~e i jv and hv i jv, resectively. Using the relation V ¼ Z I, the imedance Z is given by Z ðrþ ¼ 4 r dr 0 r 0 2 s M ðr0 Þ½ þ it ðr 0 ÞŠ ; (7) where t ðr 0 Þ¼ 3M s M and r is the distance between the center of the sorce and the osition defined by ~r. FIGURE Illstration of the two main hysical henomena involved in the genesis of local field otentials. A given crrent sorce rodces an electric field, which will tend to olarize the charged membranes arond the sorce, as schematized on the to. The flow of ions across the membrane of the sorce will also involve ionic diffsion to reeqilibrate the concentrations. This diffsion of ions will also be resonsible for indcing crrents in extracelllar sace. These two henomena inflence the freqency filtering and the genesis of LFP signals, as exlored in this article.

5 Macroscoic Model of LFPs 2593 It is, however, the most freqently sed model to calclate extracelllar field otentials (22). This model will be referred to as Model N in the following. Nonreactive media with ionic diffsion (Model D) Becase crrent sorces are ionic crrents, there is flow of ions inside or otside of the membrane, and another hysical henomena nderlying crrent flow is ionic diffsion. Let s consider a resistive medim sch as a homogeneos extracelllar condctive flid with electric arameters s m z ¼ s m ðrþð þ i3m ðrþ s m Þ¼sm þ i 3m zs m ; ðrþ s m in which the ionic diffsion coefficient is D. When the extracelllar crrent is exclsively de to ionic ffiffiffi diffsion, the crrent density deends on freqency as (see Aendices ). A resistive medim behaves as if it had a resistivity eqal to að þ b= ffiffiffi Þ, where b is comlex. The arameter a is the resistivity for very high freqencies, and reflects the fact that the effect of ionic diffsion becomes negligible comared to calorific dissiation (Ohm s law) for very high freqencies. When ionic diffsion is dominant comared to electric field effects, the real art of b is mch larger than a. The freqency deendence of condctivity will be given by s M sm ¼ : (9) þ k Alying Eq. 7 to this configration gives the following exression for the electric otential as a fnction of distance: V ð~rþ ¼, I 4s M z r ¼ þ k ffiffiffi, 4s, I m r : (20) This exression shows that, in a nonreactive medim, when the extracelllar crrent is dominated by ionic diffsion (comared to that directly rodced by the electric field), then the condctivity ffiffiffi will be freqency-deendent and will scale as. This model will be referred to as Model D in the following. Note that, if the electric field rimes over ionic diffsion, then we have the sitation described by Model N above. Reactive media with electric fields (Model P) In reality, extracelllar media contain different charge densities, for examle de to the fact that cells have a nonzero membrane otential by maintaining differences of ionic concentrations between the inside and otside of the cell. Sch charge densities will necessarily be inflenced by the electric field or by ionic diffsion. As above, we first consider the case with only electric field effects and will consider next the inflence of diffsion and the two henomena taken together. Electric olarization is a rominent tye of reaction of the extracelllar medim to the electric field. In articlar, the ionic charges accmlated over the srface of cells will migrate and olarize the cell nder the action of the electric field. It was shown reviosly in a theoretical stdy that this srface olarization henomena can have imortant effects on the roagation of local field otentials (9). If a charged membrane is laced inside an electric field ~E S 0, there is rodction of a secondary electric field ~E S given by (see Eq. 3 in (9)) ~E S ¼ ~E S 0 þ it M : (2) This exression is the freqency-domain reresentation of the effect of the inertia of charge movement associated with srface olarization, reflecting the fact that the olarization does not occr instantaneosly bt reqires a certain time to set. This freqency deendence of the secondary electric field was derived in Bédard et al. (9) for a sitation where the crrent was exclsively rodced by electric field. The arameter t M is the characteristic time for charge movement (Maxwell-Wagner time) and eqals 3 memb /s memb, where 3 memb and s memb are, resectively, the absolte (tangential) ermittivity and condctivity of the membrane srface, resectively, and are in general very different from the ermittivity and the condctivity of the extracelllar flid. Let s now determine for zero-freqency the amlitde of the secondary field ~E S 0 rodced between two cells embedded in a given electric field. First, we assme that it is always ossible to trace a continos ath which links two arbitrary oints in the extracelllar flid (see Fig. 2 B). Conseqently, the domain defined by extracelllar flid is said to be linearly connex. In this case, the electric otential arising from a crrent sorce is necessarily continos in the extracelllar flid. Second, in a first aroximation, we can consider that the celllar rocesses srronding sorces are FIGURE 2 Monoole and diole arrangements of crrent sorces. (A) Scheme of the extracelllar medim containing a qasidiole (shaded) reresenting a yramidal neron, with soma and aical dendrite arranged vertically. (B) Illstration of one of the monooles of the diole. The extracelllar sace is reresented by celllar rocesses of varios size (circles) embedded in a condctive flid. The dashed lines reresent eqiotential srfaces. The cab line illstrates the fact that the extracelllar flid is linearly connex.

6 2594 Bédard and Destexhe arranged randomly (by oosition to being reglarly strctred) and their distribtion is therefore aroximately isotroic. Conseqently, the field rodced by a given sorce in sch a medim will also be aroximately isotroic. Also conseqent to this qasirandom arrangement, the eqiotential srfaces arond a sherical sorce will necessarily ct the celllar rocesses arond the sorce (Fig. 2 B). Sose that at time t ¼ 0, an excess of charge aears at a given oint in extracelllar sace, then a static electric field is immediately rodced. At this time, crrents begins to flow in extracelllar flid, as well as inside the different celllar rocesses srronding the sorce. These cells begin to olarize, with t MW as the characteristic olarization time. Asymtotically, the system will reach an eqilibrim where the olarization will netralize the electric field, sch that there is no electric field inside the cells (zero crrent). Now sose that, in the asymtotic regime, there wold still be a crrent flowing in between cells (in the extracelllar flid); then we have two ossibilities. First, the eqiotential srfaces are discontinos, or they ct the membrane srfaces (as illstrated in Fig. 2 B). The first ossibility is imossible becase it wold imly an infinite electric field. The second ossibility is also imossible, becase cells are isootential de to olarization. Therefore, we can say that, asymtotically, there is no crrent flowing in extracelllar flid at f ¼ 0, and necessarily this is eqivalent to a dielectric medim. In other words, a assive inhomogeneos medim with randomly distribted assive cells is a erfect dielectric at zero freqency. In this case, the condctivity mst tend to 0 when freqency tends to 0. Ths, in the following, we assme that ~E S 0 ¼ ~E P 0 ~E where P 0 is the field rodced by the sorce. It follows that the exression for the crrent density in extracelllar sace as a fnction of the electric field is given by ~j ¼ s M~ z E P ¼ sm, þ i 3m, ~E P þ ~E S ¼ s m, þ i 3m, s m s m it M þ it M, ~E P : In addition, for cerebral cortical tisse, we have þ 3m s m z for freqencies >0 Hz and <000 Hz (see (8). Ths, an excellent aroximation of the condctivity can be written as Alying Eq. 7 gives V ð~rþ ¼ 4s M z s M z ¼ s m it M, : (22) þ it M, I r ¼ it M, þ it M 4s, I m r : (23) This model describes the effect of olarization in reaction to the sorce electric field, and will be referred to as Model P in the following. Reactive media with electric field and ionic diffsion (Model DP) The roagation of crrent in the medim is dominated by ionic diffsion crrents or by crrents rodced by the electric ffiffiffi field, according to the vales of k and k with resect to. The vales of k and k are, resectively, inversely roortional to the sqare root of the global ionic diffsion coefficient in the extracelllar flid, and of membrane srface (see Aendices ). We aly the reasoning based on the connex toology of the cortical medim (see above) to dedce the order of magnitde of the indced field for zero freqency ~E S 0, where ~E S ¼ ~E P 0 ; (24) þ i t t ¼ð þ k Þt M ¼ ffiffiffi 3 memb s memb; becase the tangential condctivity on membrane srface is given by s memb ¼ smemb ; þ k when the crrent is dominated by either electric field or ionic diffsion (see Eq. 9). It follows that the exression for the crrent density in extracelllar sace as a fnction of the electric field is given by ~j ¼ s M~ z E P sm ¼, þ i 3m Þ, ~E P þ ~E S þ k s m ffiffiffi z sm i t,, ~E P þ k þ i t ; becase þ i 3m s m z in cortical tisse for freqencies >0 Hz and <000 Hz (see (8)). Ths, we have the exression for the comlex condctivity of the extracelllar medim, s M z ¼ s M þ sm i t i3m ¼, ; (25) þ k þ i t where t ¼ð þ k Þt M. Ths, we have obtained a niqe exression (Eq. 25) for the aarent condctivity in extracelllar sace otside of the sorce, and its freqency deendence de to differential Ohm s law, electric olarization henomena, and ionic diffsion. These henomena are resonsible for an aarent freqency-deendence of the electric arameters, which will be comared to the freqency deendence observed in macroscoic measrements of condctivity (see Nmerical Simlation of Macroscoic Measrements, below).

7 Macroscoic Model of LFPs 2595 Finally, Eqs. 7 and 25 imly that the macroscoic imedance of a homogeneos sherical shell of width R 2 R is given by Z ¼ 4 Z R2 R r 0 2 s M dr 0 þ ¼ R 2 R, i3m 4R R 2 s M þ : i3m (26) In the following, this model will be referred as the diffsionolarization or DP model, and we will se the above exressions (Eqs. 25 and 26) to simlate exerimental measrements. Nmerical simlation of macroscoic measrements Exeriments of Gabriel et al. (8,,3) We first consider the exeriments of Gabriel et al. (8,,3), who measred the freqency deendence of electrical arameters for a large nmber of biological tisses. In these exeriments, the biological tisse was laced in between two caacitor lates, which were sed to measre the caacitance and leak crrent sing the relation I ¼ YV, imosing the same crrent amlitde at all freqencies. Becase the admittance vale is roortional to s M þ i3 M, measring the admittance rovides direct information abot s M and 3 M. To stay coherent with the formalism develoed above, we will assme that the caacitor has a sherical geometry. The exact geometry of the caacitor shold, in rincile, have no inflence on the freqency deendence of the admittance, becase the geometry will only affect the roortionality constant between s z and Y. In the case of a sherical caacitor, by alying Eq. 26, we obtain Y ¼ R þ ic ¼ 4 R R 2 R 2 R ½s M þ i3m Š ¼ 4 R R 2 R 2 R s M z : (27) We also take into accont the fact that the resistive art is always greater than the reactive art for low freqencies (<000 Hz), which is exressed by 3 M =sm : This relation can be verified, for examle, from the measrement of Gabriel et al. (8), where it is tre for the whole freqency band investigated exerimentally (between 0 and 0 0 Hz). The real art of s M ¼ s z then takes the form ffiffiffi s M z sm t 2, þ k t 2 þ ; (28) where t ¼ð þ k Þt M. By sbstitting this vale of t, the inverse of the condctivity (the resistivity) is given by z s, þ k, þ M t 2 M ð þ k Þ 2 ¼ s, þ k k þ þ M t 2 M 3=2 t 2 M : þ 2k þ k 2 (29) s M Finally, to rerodce the exeriments of Gabriel et al., we assme k [ ffiffiffi. By develoing in series the last term (in arentheses) of Eq. 29, we obtain s M zk 0 þ K =2 þ K 2 þ K 3 3=2 ¼ K 0 þ K f þ K 2 =2 f þ K 3 f 3=2: (30) Eqation 30 corresonds to the condctivity s M, as measred in the exerimental conditions of the exeriments of Gabriel et al. (the ermittivity 3 M is obtained by alying Kramers-Kronig relations). Fig. 3 shows that this exression for the condctivity can exlain the measrements in the freqency range of Hz, which are relevant for LFPs. To obtain this agreement, we had to assme in Eq. 25 a relatively low Maxwell-Wagner time of ~0.5 s (f c ¼ /(2t M ) between Hz and 0 Hz), k > ffiffiffi > k (for freqencies <00 Hz). This vale of Maxwell-Wagner time is necessary to exlain Gabriel s exeriments, and may seem very large at first sight. However, the Maxwell-Wagner time is not limited by hysical constraints, becase we have by definition t MW ¼ 3 s. In rincile, the vale of s can be very small, aroaching zero, while 3 can take very large vales. For examle, taking the measrements of Gabriel et al. in aqeos soltions of NaCl and in gray matter (8), gives vales of t MW comrised between ms and 00 ms for freqencies at ~0 Hz. Ths, the model redicts that in the exeriments of Gabriel et al., the transformation of electric crrent carried by electrons to ionic crrent in the biological medim necessarily imlies an accmlation of ions at the lates of the caacitor. This ion accmlation will in general deend on freqency, becase the condctivity and ermittivity of the biological medim are freqency-deendent. This will create a concentration gradient across the biological medim, which will case a ionic diffsion crrent oosite to the electric crrent. This ionic crrent will allow a greater reslting crrent becase srface olarization is oosite to the electric field. Fig. 3 shows that sch conditions give freqency-deendent macroscoic arameters consistent with the measrements of Gabriel et al. The arameter choices to obtain this agreement can be jstified qalitatively becase the ionic diffsion constant on celllar srfaces is robably mch smaller than in the extracelllar flid, sch that k [ k. This imlies the existence of

8 2596 Bédard and Destexhe low freqencies, nlike this model, which is entirely dedced from well-defined hysical henomena. Logothetis et al. (0) measrements FIGURE 3 Models of macroscoic extracelllar condctivity comared to exerimental measrements in cerebral cortex. The exerimental data (labeled G) show the real art of the condctivity measred in cortical tisse by the exeriments of Gabriel et al. (8). The crve labeled E reresents the macroscoic condctivity calclated according to the effects of electric field in a nonreactive medim. The crve labeled D is the macroscoic condctivity de to ionic diffsion in a nonreactive medim. The crve labeled P shows the macroscoic condctivity calclated from a reactive medim with electric-field effects (olarization henomena). The crve labeled DP shows the macroscoic condctivity in the fll model, combining the effects of electric olarization and ionic diffsion. Every model was fit to the exerimental data by sing a least-sqare rocedre, and the best fit is shown. The DP model s condctivity is given by Eq. 30 with K 0 ¼ 0.84, K ¼ 9.29, K 2 ¼ 80.35, and K 3 ¼ The exerimental data (G) is the arametric Cole-Cole model (2), which was fit to the exerimental measrements of Gabriel et al. (8). This fit is in agreement with exerimental measrements for freqencies >0 Hz. No exerimental measrements exist for freqencies <0 Hz, and the different crves show different redictions from the henomenological model of Cole-Cole and these models. a freqency band B f for which ffiffiffi is negligible with resect to k, bt not with resect to k becase these constants are inversely roortional to the sqare root of their resective diffsion coefficients. Ths, the aroximation that we sggest here is that this band B f finishes at ~00 Hz in the exerimental conditions of Gabriel et al. It is imortant to note that this arameter choice is entirely deendent on the ratio between ionic diffsion crrent and the crrent rodced by the electric field, and ths will be deend on the articlar exerimental conditions. It is interesting to note that this model and the henomenological Cole-Cole model (2) redict different behaviors of the condctivity for low freqencies (<0 Hz). In this model, the condctivity tends to zero when freqency tends to zero, while in the Cole-Cole extraolation, it tends to a constant vale (3). The main difference between these models is that the Cole-Cole model is henomenological and has never been dedced from hysical rinciles for We next consider the exeriments of Logothetis et al. (0), which reorted a resistive medim, in contrast with the exeriments of Gabriel et al. In these exeriments, for electrodes were aligned and saced by 3 mm in monkey cortex. The first and last electrodes were sed to inject crrent, while the two intermediate electrodes were sed to measre the extracelllar voltage. The voltage was measred at different freqencies and crrent intensities. One imortant oint in this exerimental set is that the intensity of the crrent was sch that the voltage at the extreme (injecting) electrodes satrates. One of the conseqences of this satration was to limit ionic diffsion effects, as discssed in Logothetis et al. (0). This voltage satration will diminish the concentration difference near the sorce (we wold have an amlification if this was not the case). It follows that, in the exeriments of Logothetis et al., the ratio between diffsion crrent and electric field crrent is very small. Ths, in this case, we se vales of arameters k i very different from those assmed above to rerodce the exeriments of Gabriel et al., in articlar k ffiffiffi. As we will see in Discssion, this sitation may be different from the hysiological conditions. Nevertheless, the large distance between electrodes sggests that the relation between crrent and voltage is linear becase the crrent density is roghly roortional to the inverse of sqared distance to the sorce. Conseqently, we can sose that in the exeriments of Logothetis et al., the ionic gradient is negligible, which revents ionic diffsion crrents. Ths, in this exeriment, most if not all of the extracelllar crrent is de to electric-field effects. In this sitation, the condctivity (Eq. 25) becomes s M ðt M Þ 2 zsm, 2; (3) þðt M Þ which is similar to the Model P above. Moreover, taking the same Maxwell-Wagner time t M as above for the exeriments of Gabriel et al. (which corresonds to a ctoff freqency of Hz), we have for freqencies >0 Hz, ðt M Þ 2 þðt M Þ 2z; similar to a resistive medim. Ths, in the exeriments of Logothetis et al., the satration henomenon entrains crrent roagation in the biological medim as if the medim was qasiresistive for freqencies >0 Hz. This constittes a ossible exlanation of the contrasting reslts in the measrements of Logothetis et al. and Gabriel et al.

9 Macroscoic Model of LFPs 2597 Freqency deendence of the ower sectral density of extracelllar otentials The third tye of exerimental observation is the fact that the ower sectral density (PSD) of LFPs or EEG signals dislays /f freqency scaling ( 4). To examine whether this /f scaling can be acconted for by this formalism, we consider a sherical crrent sorce embedded in a continos macroscoic medim. We also assme that the PSD of the crrent sorce is a Lorentzian, which cold derive, for examle, from randomly occrring exonentially decaying ostsynatic crrents (2) (see Fig. 4). To simlate this sitation, we sed the diffsion-olarization model with ionic diffsion and electric field effects in a reactive medim. We have estimated above that srface olarization henomena have a ctoff freqency of ~ Hz, and will not lay a role above that freqency. So, if we focs on the PSD of extracelllar otentials in the freqency range > Hz,we can consider only the effect of ionic diffsion (in agreement with the exeriments of Gabriel et al.; see above). Ths, we can aroximate the condctivity as (see Eq. 25) s M ¼ a ffiffiffi ; (32) where a is a constant. It follows that the extracelllar voltage arond a sherical crrent sorce is given by (see Eq. 7) I Vðr; Þ ¼ 4a ffiffiffi Vðr; Þ VðR; ÞR ¼ ¼ ffiffiffi ; (33) r r where R is the radis of the sorce. In other words, we can say that the extracelllar otential is given by the crrent sorce I convolved with a filter in = ffiffiffi, which is essentially de to ionic diffsion (Warbrg imedance; see the literatre (23 25).). A white noise crrent sorce will ths reslt in a PSD scaling as /f, and can exlain the exerimental observations, as shown in Fig. 4. Exerimentally recorded LFPs in cat arietal cortex dislay LFPs with freqency scaling as /f for low freqencies, and /f 3 for high freqencies (Fig. 4, A and B). Following the same rocedre as in Bédard et al. (2), we reconstrcted the synatic crrent sorce from exerimentally recorded sike trains (Fig. 4, C and D). The PSD of the crrent sorce scales as a Lorentzian (Fig. 4 E) as exected from the exonential natre of synatic crrents. Calclating the LFP arond the sorce and taking into accont ionic diffsion, gives a PSD with two freqency bands, scaling in /f for low freqencies, and /f 3 for high freqencies (Fig. 4 F). This is the freqency scaling observed exerimentally for LFPs in awake cat cortex (2). We conclde that ionic diffsion is a lasible hysical case of the /f strctre of LFPs for low freqencies. A C B E D F FIGURE 4 Simlation of /f freqency scaling of LFPs dring wakeflness. (A) LFP recording in the arietal cortex of an awake cat. (B) Power sectral density (PSD) of the LFP in log scale, showing two different scaling regions with a sloe of and 3, resectively. (C) Raster of eight simltaneosly-recorded nerons in the same exeriment as in anel A. (D) Synatic crrent calclated by convolving the sike trains in anel C with exonentials (decay time constant of 0 ms). (E) PSD calclated from the synatic crrent, shown two scaling regions of sloe 0 and 2, resectively. (F) PSD calclated sing a model inclding ionic diffsion (see text for details). The scaling regions are of sloe and 3, resectively, as in the exeriments in anel B. Exerimental data taken from Destexhe et al. (37); see also Bédard et al. (2) for details of the analysis in anels B D.

10 2598 Bédard and Destexhe Two imortant oints mst be noted. First, the diffsionolarization model does not atomatically redict /f scaling at low freqencies, bt rather imlements an /f filter, which may reslt in freqency scaling with larger sloes. Second, the same exerimental sitation may reslt in different freqency scaling, and this is also consistent with the diffsion-olarization model. These two oints are illstrated in Fig. 5, which shows a similar analysis as Fig. 4 bt dring slow-wave slee in the same exeriment. The LFP is dominated by slow-wave activity (Fig. 5 A), and the different nits dislay firing atterns characterized by concerted ases (shaded lines in Fig. 5 B), characteristic of slow-wave slee and which are also visible in the reconstrcted synatic crrent (Fig. 5 C). The PSD shows a similar scaling as /f 3 as for wakeflness, bt the scaling at low freqencies is different (sloe at ~ 2 at low freqencies; see Fig. 5 D). The PSD reconstrcted sing the diffsion-olarization model dislays similar featres (comare with Fig. 5 E). This analysis shows that the diffsion-olarization model qalitatively acconts for different regions of freqency scaling fond exerimentally in different freqency bands and network states. Measrement of freqency deendence In this final section, we examine a ossible way to test the model exerimentally. The main rediction of the model is that, in natral conditions, the extracelllar crrent A B C D E FIGURE 5 Simlation of more comlex freqency scaling of LFPs dring slow-wave slee. (A) Similar LFP recording as in Fig. 4 A (same exeriment), bt dring slow-wave slee. (B) Raster of eight simltaneosly-recorded nerons in the same exeriment as in anel A. The vertical shaded lines indicate concerted ases of firing which resmably occr dring the down states. (C) Synatic crrent calclated by convolving the sike trains in anel B with exonentials (decay time constant of 0 ms). (D) Power sectral density (PSD) of the LFP in log scale, showing the same scaling regions with a sloe of 3 at high freqencies as in wakeflness (the PSD in wake is shown in shading in the backgrond). At low freqencies, the scaling was close to /f 2 (shaded line; the dotted line shows the /f scaling of wakeflness). (E) PSD calclated from the synatic crrent in anel C, sing a model inclding ionic diffsion. This PSD rerodces the scaling regions of sloe 2 and 3, resectively (shaded lines). The low-freqency region, which was scaling as /f in wakeflness (dotted lines), had a sloe close to 2. Exerimental data taken from Destexhe et al. (37).

11 Macroscoic Model of LFPs 2599 erendiclar to the sorce is dominated by ionic diffsion. The exeriments realized so far (8,0,,3) sed macroscoic crrents that did not necessarily resect the correct crrent flow in the tisse. We sggest creating more natralistic crrent sorces by generating ionic crrents with a microiette laced in the extracelllar medim. By sing eriodic crrent injection dring very short time Dt comared to the eriod (small dty cycle), we can measre, sing the same electrode, the extracelllar voltage V r (sing a fixed reference far away from the sorce). If the eriod of the sorce is shorter than the relaxation time of the system, the voltage will integrate, which is de to charge accmlation. Becase V r is directly roortional to the amont of charge emitted as a fnction of time dring Dt (caacitive effect of the extracelllar medim), the time variation of V r is directly roortional to the ionic diffsion crrent. In sch conditions, if the extracelllar medim is rely resistive as redicted by the exeriments of Logothetis et al., the relaxation time shold be very small, of ~0 2 s(9), which wold revent any integration henomena and charge accmlation for freqencies <0 2 Hz. If the medim has a slower relaxation de to olarization and ionic diffsion, then we shold observe voltage integration and charge accmlation for hysiological freqencies (<000 Hz). To illstrate the difference between these two sitations, we consider the simlest case of a nonreactive medim (as in Model D above), in which the crrent can be rodced by ionic diffsion or by the electric field, or by both. To calclate the time variations of ionic concentration and extracelllar voltage, we consider the crrent density: ~j ¼ DVe½cŠþs~E: (34) According to the differential law for charge conservation and Poisson law, we have V~j þ vr vt ¼ D V 2 r þ s 3 r þ vr vt ¼ 0: (35) When ionic diffsion is negligible comared to Ohm s law, we have vr vt þ s 3 rz0: It follows that the charge density is given by r ¼ r o ex s 3 t : (36) On the other hand, if ionic diffsion is the rimary case of crrent roagation in the extracelllar medim, then the relaxation time shold be mch larger and ths, integration shold be observed. When ionic diffsion is dominant, we have instead of Eq. 35. vr vt þ D V2 rz0 The general soltion is r ¼ ffiffiffiffi 2Dt 0 rðr; 0Þe r2 4Dt dr: (37) The difference between the exressions above (Eqs. 36 and 37) shows that the time variation of charge density is different according to which crrent dominates, electric field crrent or ionic diffsion crrent. The same alies to the electric otential between the electrode and a given reference, becase this otential is linked to charge density throgh Poisson s law. Therefore, this exeriment wold be crcial to clearly show which of the two crrents rimes for crrents erendiclar to the sorce (this wold not aly to longitdinal crrents, like axial crrents in dendrites). In the hyothetical case of dominant ionic diffsion, the cortex wold be similar to a Warbrg imedance and one can estimate the macroscoic diffsion coefficient sing Eq. 37. Ths, sing a microiette injecting eriodic crrent lses, it shold be ossible to test the caacity of the medim to create charge accmlation for hysiological freqencies. If this is the case, it wold constitte evidence that ionic crrents are nonnegligible in the hysiological sitation. DISCUSSION In this article, we have roosed a framework to model local field otentials, and which synthesizes revios measrements and models. This framework integrates microscoic measrements of electric arameters (condctivity s and ermittivity 3) of extracelllar flids, with macroscoic measrements of those arameters (s M, 3 M ) in cortical tisse (8,0). It also integrates revios models of LFPs, sch as the continm model (7), which was based on a continm hyothesis of electric arameters variations in extracelllar sace, or the olarization model (9), which exlicitly considered different media (flid and membranes) and their olarization by the crrent sorces. This model is more general and also integrates ionic diffsion, which is redicted as a major determinant of the freqency deendence of LFPs. This diffsion-olarization model also acconts for observations of /f freqency scaling of LFP ower sectra, which is de here to ionic diffsion, and is therefore redicted to be a conseqence of the genesis of the LFP signal, rather than being solely de to neronal activity (see (2)). Finally, this work sggests that emhatic interactions between nerons can occr not only throgh electric fields bt that ionic diffsion shold also be considered in sch interactions. As discssed in Simlified Geometry for Macroscoic Parameters, this model rests on several aroximations, which were necessary to obtain the analytic exressions sed here. These aroximations were that crrent sorces were considered as monoolar entities (longitdinal crrents

12 2600 Bédard and Destexhe sch as axial crrents in dendrites were not taken into accont), the crrent sorce was sherical and the extracelllar medim was considered isotroic. Becase mltiolar effects can be reconstrcted from the serosition of monooles (20), the monoolar configration shold not affect the reslts on freqency deendence, as long as the extracelllar crrent erendiclar to the sorce is considered. Similarly, the exact geometry of the crrent sorce shold have no inflence on the freqency deendence far away from the sorces. However, in the immediate vicinity of the sorces, the geometric natre and the synchrony of synatic crrents can have inflences on the ower sectrm (26). Another assmtion is that the extracelllar medim is isotroic, which was jstified within the macroscoic framework followed here. These factors, however, will inflence the exact shae of the LFPs. More qantitative models inclding a more sohisticated geometry of crrent sorces and the resence of membrane excitability and action otentials shold be considered (e.g., (26,27)). The main rediction of this model is that ionic diffsion is an essential hysical case for the freqency deendence of LFPs. We have shown that the resence of ionic diffsion allows the model to accont qantitatively for the macroscoic measrements of the freqency deendence of electric arameters in cortical tisse (8). Ionic diffsion is resonsible for a freqency deendence of the imedance as = ffiffiffi for low freqencies (<000 Hz), which directly acconts for the observed /f freqency scaling of LFP and EEG ower sectra dring wakeflness ( 4) (see Fig. 4). Note that the EEG is more comlex becase it deends on the diffsion of electric signals across flids, dra matter, skll, mscles, and skin. However, this filtering is of lowass tye, and may not affect the low-freqency band, so there is a ossibility that the /f scaling of EEG and LFPs have a common origin. This model is consistent with the view that this aarent /f noise in brain signals is not generated by self-organized featres of brain activity, bt is rather a conseqence of the genesis of the signal and its roagation throgh extracelllar sace (2). It is imortant to note that the fact that ionic diffsion may be resonsible for /f freqency scaling of LFPs is not inconsistent with other factors, which may also inflence freqency scaling. For examle, the statistics of network activity and more generally network state can affect freqency scaling. This is aarent when comaring awake and slow-wave slee LFP recordings in the same exeriment, showing that the /f scaling is only seen in wakeflness bt /f 2 scaling is instead seen dring slee (2) (see Fig. 5). In agreement with this, recent reslts indicate that the correlation strctre of synatic activity may inflence freqency scaling at the level of the membrane otential, and that correlated network states scale with larger (more negative) exonents (28). We also investigated ways to exlain the measrements of Logothetis et al. (0), who reorted that the extracelllar medim was resistive and therefore did not dislay freqency deendence, in contradiction with the measrements of Gabriel et al. (8). We smmarize and discss or conclsions below. In the exeriments of Gabriel et al. (8), one measres ermittivity and condctivity in the medim in between two metal lates. This forms a caacitor, which (macroscoic) comlex imedance is measred. This measre actally consists of two indeendent measrements, the real and imaginary art of the imedance. These vales are sed to dedce the macroscoic ermittivity and macroscoic condctivity of the medim. However, at the interface between the medim and the metal lates, the flow of electrons in the metal corresonds to a flow of charges in the tisse, and a variety of henomena can occr, which can interfere with the measrement. The accmlation of charges that occrs at the interface between the electrode and the extracelllar flid imlies a olarization imedance, which deends on the interaction between ions and the metal late. Becase this accmlation of charge imlies a variation of concentration, the flow of ions may involve an imortant comonent of ionic diffsion. In the exeriments of Logothetis et al., a system of for electrodes is sed; the two extreme electrodes inject crrent in the medim, while the two electrodes in the middle are exclsively sed to measre the voltage. This system is sosed to be more accrate than Gabriel s, becase the electrodes that measre voltage are not sbject to charge accmlation. However, the drawback of this method are nonlinear effects. The magnitde of the injected crrent is sch that the voltage at the extreme electrodes satrates. This voltage satration also imlies satration of concentration (caacitive effect between electrodes), which limits ionic diffsion crrents. Ths, the ratio between ionic diffsion crrents and the crrents de to the electric field is greatly diminished relative to the exeriments of exeriments of Gabriel et al. We think that natral crrent sorces are closer to the sitation of Gabriel et al. for several reasons. First, the magnitde of the crrents rodced by biological sorces is far too low for satration effects. Second, the flow of charges across ion channels will rodce ertrbations of ionic concentration, which will be reeqilibrated by diffsion. The effects may not be as strong as the ertrbations of concentrations indced by the exeriments of Gabriel et al., bt ionic diffsion shold lay a role in both cases. This is recisely one of the asects that shold be evalated in frther exeriments. The exeriments of Logothetis et al. were done sing a for-electrode set, which netralizes the inflence of electrode imedance on voltage measrements (29,30). This system was sed to erform high-recision imedance measrements, also avoiding ionic diffsion effects (0). Indeed, these exerimental conditions, and the aarent resistive medim, cold be rerodced by this model if ionic diffsion was neglected. This model therefore formlates the strong rediction that ionic diffsion is imortant, and that