Modeling the Link between Functional Imaging and Neuronal Activity: Synaptic Metabolic Demand and Spike Rates

NeuroImage 17, 1065–1079 (2002) doi:10.1006/nimg.2002.1234

Modeling the Link between Functional Imaging and Neuronal Activity: Synaptic Metabolic Demand and Spike Rates Rita Almeida* ,† ,1 and Martin Stetter‡ *Division of Human Brain Research, Karolinska Institute, 17177 Stockholm, Sweden; †Institute of Biophysics and Biomedical Engineering, University of Lisbon, Lisbon, Portugal; and ‡Corporate Technology, Information and Communications, Siemens AG, 81730 Munich, Germany Received February 19, 2002

Functional magnetic resonance imaging (fMRI) and positron emission tomography (PET) measurements reflect changes in the hemodynamics which are thought to be related to local synaptic input to neuron populations. The local neuronal spiking activity, which is believed to form the basis of neuronal coding and communication, is not directly reflected in fMRI/ PET measurements. We used a mean-field neuronal model of recurrently coupled excitatory and inhibitory neuronal populations to characterize the relationship between the synaptic activity (reflected in the PET and fMRI measurements) and the neuronal spike rates, averaged over brain areas. We analyzed this relation for a number of cases. For a single brain area and in the absence of external input to its inhibitory neurons, the relation between average spike rates and synaptic activity is linear. However, departures from linearity are found when: (i) the local synaptic strengths vary, (ii) the external inputs vary, in the presence of external input to the inhibitory population, or (iii) the synchronization between oscillations of the average spike rates of two areas changes. We further show that an increase in the imaging signal can reflect a decrease in average spiking activity, in the presence of external input to the inhibitory population. Synaptic activity can also be associated with silent neuronal populations, when input to the excitatory population does not reach the activation threshold or for certain synchronizations between oscillations of two areas. In conclusion, caution should be used when interpreting neuroimaging results in terms of mean spike rates. © 2002 Elsevier Science (USA)

INTRODUCTION During the past decades, positron emission tomography (PET) and functional magnetic resonance imaging 1

To whom correspondence and reprint requests should be addressed. E-mail: [email protected].

(fMRI) have been widely established as techniques for monitoring task-related brain activity in humans. Despite the effort invested in understanding the physiological basis of neuroimaging, the relation between imaging measurements and neuronal activity is still not fully understood (for reviews see, for example, Jueptner and Weiller, 1995; Villringer and Dirnagl, 1995; Magistretti and Pellerin, 1996, 1999; Villringer, 1999). PET is commonly used to measure relative changes in local blood flow, found to be correlated with changes in metabolic consumption associated with neuronal activity. The most used fMRI signal, blood oxygenation level-dependent (BOLD) signal, is also thought to indirectly reflect changes in metabolic demand, related to neuronal activity. The BOLD signal depends on local blood volume, flow, and level of blood oxygenation, but the exact hemodynamic mechanisms coupling BOLD to neuronal activity remain unknown. Local cerebral metabolic changes were shown to more closely reflect synaptic activity than spiking activity (Kadekaro et al., 1985; Nudo and Masterton, 1986). Further, inhibitory synapses have been also associated with increased metabolic demand (Nudo and Masterton, 1986). Whether excitatory or both excitatory and inhibitory synapses contribute to the neuroimaging signals is a matter of debate. A recent fMRI study in humans attributes the origin of the BOLD signal mostly to synaptic excitation (Waldvogel et al., 2000), which seems to be in agreement with the higher density of excitatory synapses in brain tissue. Whether PET and BOLD signals correlate better with synaptic than with spiking activity remains controversial. The controversy may originate in the complexity of the relationship between synaptic and spiking activities, which the present modeling work addresses and attempts to clarify. Evidence pointing in this direction came from a recent work by Logothetis and co-workers (Logothetis et al., 2001), who directly examined the relationship between BOLD signal and neuronal activity, by simulta-

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neous measuring local field potentials (LFP), multiunit activity, and BOLD signal on the monkey visual cortex. The LFP were a better predictor of the BOLD signal than the multiunit activity. A fraction of neurons showed fast adapting transient spiking behavior, which was reflected neither in the sustained LFP nor in BOLD signal. Since LFP is mainly associated with synaptic activity the results indicate that this quantity, rather than spike rates, is reflected in BOLD signal. Although LFP were significantly better predictors of BOLD signal, the multiunit activity also tended to be a good predictor. Further, action potentials also contribute to LFP and hence conclusions about the processes reflected in BOLD should be treated with caution. For a discussion see Bandettini and Ungerleider (2001). Local information processing within a brain area is based on the exchange of spike trains through local recurrent circuits, and information between brain areas is communicated via spike patterns carried by pyramidal neurons. However, fMRI and PET signals are not only measuring the spiking activity of local neuronal populations, but they can also be measuring subthreshold excitatory and inhibitory activities and modulatory inputs from other areas. Therefore, to interpret functional imaging data in terms of neural computation it is important to understand the relationship between the summed synaptic activity and the corresponding population spiking activity. In fact, contradictory results have been found with electrophysiology and neuroimaging. Using electrophysiology, attentional effects in area V1 of the monkey brain are very difficult to measure, while they have been reported in humans, using fMRI (Heeger and Ress, 2002). The attentional effect measured with fMRI could correspond to subthresholded activity not evoking spiking activity. On the other hand, recent comparisons between human fMRI and single-unit recordings in monkey have hinted at an approximately linear relationship between the single-unit spike rate and the BOLD contrast in V5 (Rees et al., 2000) and V1 (Heeger et al., 2000), although the constants of proportionality seen in both studies differed from each other about 20-fold. In general, however, synaptic input and spiking output cannot be considered related in a simple way: Neurons are nonlinear units, and the spiking output of recurrently coupled populations of neurons is determined from the synaptic input by the nonlinear recurrent dynamics of these populations. Synaptic input and spiking output can also differ qualitatively from each other, if neurons are strongly tuned in feature space or time and if they encode information by the deviation of their firing from the mean firing rate. Scannell and Young (1999) explored this relationship between population spiking activity and single neuron firing. They showed, that the BOLD signal can be insensitive to the spiking activity of strongly

tuned neurons, which represent important information about the input, because their activity is eventually masked by small changes in the baseline of the overall mean firing rate. Furthermore the mean spike rate of a neuron pool is not necessarily related to its summed synaptic input in a linear way. This issue has been theoretically addressed recently (Tagamets and Horwitz, 2001) and is extended in the present work. The relationship between total synaptic input and mean spike rate averaged over brain areas was modeled using a mean-field formulation of the recurrent dynamics at the level of neuron populations. Situations where the relation between spiking and synaptic activities is linear and where a departure from linearity is found were characterized. We found that in the absence of external input to the inhibitory neurons, the modeled relationship is constant in function of the external input, but this constant is dependent on the local synaptic strengths. When external input to the inhibitory neurons is considered, the relationship between spiking and synaptic activities is a nonlinear function of the external inputs. The relationship investigated also depends nonlinearly on the threshold for neuronal activation and on the synchrony between oscillatory activities of connected areas. The results are relevant for any technique measuring mainly synaptic activity with low spatial and temporal resolutions, such as optical imaging. In the Discussion the results are presented in the context of neuroimaging, and the average synaptic activity is interpreted as reflecting PET and fMRI measurements. METHODS Here we formulate a mean-field neurodynamical model for the time evolution of the average synaptic activity and the mean spiking activity in a system of recurrently connected brain areas. Mean-Field Dynamics of Connected Brain Areas Mean-field models (Wilson and Cowan, 1973; BenYishai et al., 1995; Gerstner, 2000; Stetter, 2001) describe the collective spike dynamics of a large number of neurons. They are based on the assumption that the neurons considered receive similar input signals and as a consequence behave similarly on average. For such a pool of neurons, only the mean behavior is described, whereas the fluctuations in the system are omitted. Figure 1 illustrates the scenario for which we set up a model. Let us assume that under a certain task we measure a nonzero signal reflecting averaged synaptic activity (using, for example, PET or fMRI) in several brain areas of a subject, including the two regions marked “area 1” and “area 2” in the figure. We aim at

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tions between the areas originate only from the excitatory pool, because they reflect interareal connections, which are established by pyramidal neurons. The mean strength of synapses from area ␣ ⫽ 1, 2 to cell type a ⫽ e, i of the other area is denoted by L a␣. Finally, each pool receives external input h a␣, which generically summarizes afferents from other brain areas which are not explicitly modeled. The mean input received by a neuron pool is modeled as a sequence of excitatory (EPSP) and inhibitory (IPSP) postsynaptic potentials with fixed time courses. Because of their extension over time, the mean input to a pool at present time t will generally be affected by the activity the presynaptic neuron pool had in the past. In Appendix 1 we derive a simple time-resolved meanfield framework. We show that the mean local input to, say, the pool of excitatory neurons in area 1 from itself can be written as

H ee共t兲 ⫽

冕

⬁

S e␧e共t⬘兲me1共t ⫺ t⬘兲dt⬘ ⫽ Se␧e ⴱ me1 ,

(1)

0

FIG. 1. Architecture of a mean-field neurodynamical model for the metabolic demand and the spiking activity within a set of two mutually connected brain regions; e’s designate pools of excitatory neurons and i’s pools of inhibitory neurons; h e and h i are inputs external to the represented areas to the excitatory and inhibitory populations, respectively; S’s are weights of local connections, while L’s are weights of connections between areas. S e and L e are weights of connections originating on excitatory populations. S i and L i are weights of connections originating on inhibitory populations.

a model which allows us to predict the mean spiking activity in each of these brain areas for a given level of synaptic activity. Here we identify the brain regions with cortical areas, but the model can be used for other brain structures. We assume that the two areas under consideration receive connections from each other and also receive input from other brain areas and/or sensory systems. Additionally, each area maintains some local recurrent circuitry, which consists of both excitatory synapses from pyramidal neurons and inhibitory synapses from inhibitory interneurons, such as cortical stellate cells. Next we identify the set of all pyramidal (excitatory) neurons of one area with one population labeled “e” and the set of all inhibitory cells with another pool labeled “i,” arriving at two populations per area (Fig. 1, bottom). The two populations of each area are locally interconnected by recurrent mean connections with strengths S e and S i, respectively. The arrangement shown assumes that the axons do not prefer contacting one cell type over the other: Each neuron pool maintains the same synaptic connection strength to itself and to the respective other pool. Long-range connec-

where ␧ e(t) denotes the time course of an EPSP and m e1(t) describes the mean spiking activity of the neuron pool e1, i.e., the fraction of neurons in the pool which fire a spike at a small time interval around t. The second partof the equation introduces a shorthand notation for the first part. The mean synaptic input to all four areas becomes H e1 ⫽ Se␧e ⴱ me1 ⫺ Si␧i ⴱ mi1 ⫹ Le2␧e ⴱ me2 ⫹ he1 ,

(2)

H i1 ⫽ Se␧e ⴱ me1 ⫺ S i␧i ⴱ mi1 ⫹ Li2␧i ⴱ me2 ⫹ hi1 ,

(3)

H e2 ⫽ Se␧e ⴱ me2 ⫺ S i␧i ⴱ mi2 ⫹ Le1␧e ⴱ me1 ⫹ he2 ,

(4)

H i2 ⫽ Se␧e ⴱ me2 ⫺ Si␧i ⴱ mi2 ⫹ Li1␧i ⴱ me1 ⫹ hi2 ,

(5)

where the explicit notation of time has been omitted for simplicity. Because the model incorporates the explicit time courses of the synaptic potentials, it can reproduce oscillatory spiking behavior in a natural way. The mean spiking activity of a pool (a␣), denoted by m a␣(t), increases if the pool as a whole receives depolarizing inputs carried by EPSPs, and it decreases if it receives hyperpolarizing input carried by IPSPs. In Appendix 2 we provide the framework for the dynamics of the mean-field model and show that the time evolution of the mean spiking activity can be approximated by

␶

d dt

ma␣ ⫽ ⫺ma␣ ⫹ g a共H a␣兲,

a ⫽ e, i,

␣ ⫽1, 2 ,

(6)

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where g a(H) is an activation function which is zero below a threshold T and linearly increasing above the threshold.

replaced by its linear part. The fixed point solutions M a␣ for the four mean activations then obey

冢冣冢 M e1

Mean Synaptic Activity in Brain Areas

M i1

We aimed at modeling the relationship between average spiking activity of a neuronal population and its corresponding total synaptic activity. The total synaptic activity is taken to be a weighted sum of the absolute values of the inputs to that area. This definition allows synaptic activity to be related to neuroimaging signals, as will be discussed later. The instantaneous values of the synaptic activities of areas 1 and 2 are given by U1 ⫽ U e0 共2Se␧e ⴱ me1 ⫹ 共Le2␧e ⫹ Li2␧i兲 ⴱme2 ⫹ he1 ⫹ hi1) ⫹ 2Ui0Si␧i ⴱ mi1 , U 2 ⫽ U e0共2S e␧ e ⴱ m e2 ⫹ 共L e1␧ e ⫹ L i1␧ i兲 ⴱ m e1 ⫹ h e2 ⫹ h i2) ⫹ 2U i0S i␧ i ⴱ m i2

(7)

M e2 M i2

M

⫽

S e ⫺Si L e

0

S e ⫺Si L i

0

冣冢 冣 冢 冣 h e1 ⫺ T

M e1 M i1

Le

0

S e ⫺Si

Li

0

S e ⫺Si

M e2

⫹

M i2

h i1 ⫺ T

h e2 ⫺ T h i2 ⫺ T

h⫺T

W

,

(9)

and they can be written as M ⫽ 共I ⫺ W兲 ⫺1 共h ⫺ T兲 ,

(10)

if the matrix I ⫺ W is invertible (I is the 4 ⫻ 4 identity matrix). Numerical Simulations

(8)

for brain areas 1 and 2, respectively. The factors U e0 and U i0 weigh the contributions of excitatory and inhibitory synaptic transmissions relative to each other. These equations indicate the following major potential differences between synaptic and spiking activities. First, synaptic potentials can be evoked by active neurons from remote sites, e.g., for spikes propagating between areas. Hence synaptic and spiking activities can be spatially distant. Second, synaptic activity is proportional to the absolute sum of synaptic potentials, whereas spiking activity forms as a nonlinear function of their direct sum: Both quantities generally can show a nonlinear and complex dependence on each other. Third, for the same number of incoming spikes (the same energy consumption), differences in their time courses, for example, during different collective activity states such as oscillations, can lead to different levels of spiking activity. Under Results we characterize some of the conditions where synaptic activity reliably reflects spiking activity or when there are deviations to be expected. Solution for Stationary Superthreshold Input For the case where all neuron populations receive suprathreshold input and where Eqs. (6) evolve into a time-independent fixed-point solution, an analytical solution for the steady-state activations can be provided. After transients, the time derivative on the lefthand side of Eq. (6) vanishes, and the convolutions on the righthand side can be integrated out analytically. Because all inputs are assumed suprathreshold, the activation function g on the righthand side can be

The results in the next section were obtained using Eq. (10) where possible; otherwise Eqs. (1)–(6) were integrated numerically. The synaptic activity was calculated using Eqs. (7) and (8). Simulations were carried out using Matlab 6.1 on SUN Ultra 5 workstations and took approximately 5 s for each second of simulation. For the sake of simplicity, one single brain area receiving external input was used to study how different external inputs to an area and connection strengths affect the relation between spike rates and synaptic activity. Figure 2a illustrates the model considered for one brain area. Two connected brain areas were used to study the combined effect of different connectivity between areas and different dynamical states of the system. Time averages of the quantities of interest over several seconds were considered, because the time resolution of fMRI and PET is on the order of seconds, and so variations on a smaller time scale cannot be detected by these techniques and thus are outside the scope of this work. The parameters of the activation function (g(H) ⫽ max(␤(H ⫺ T), 0)) considered were ␤ ⫽ 1 and T ⫽ 0, and we used a time constant of ␶ ⫽ 1 ms for the differential Eq. (6). The functions ␧ a(t), a ⫽ e, i, for the shape of the PSPs we used in this work are of the form ␧ a共t兲 ⫽

t

␶a

e ⫺共t/␶a兲 ,

(11)

where ␶ a is a time constant characteristic of the excitatory or inhibitory synapse. We considered ␶ e ⫽ ␶ i ⫽ 5 ms, but all the results presented except the frequency of synchronous oscillation are independent of these

SYNAPTIC ACTIVITY AND SPIKE RATES

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Effect of Local Connection Strength

FIG. 2. (a) One brain area modeled as two interconnected pools of neurons, one of excitatory and one of inhibitory cells. (b) Model of two connected brain areas for the special case of a feed-forward connection.

values (the stationary solutions do not depend on these parameters). The synaptic activity can be calculated considering the contributions of both excitatory and inhibitory synapses (U e0 ⫽ U i0 ⫽ 1) or only of excitatory synapses (U e0 ⫽ 1, U i0 ⫽ 0). Results for both situations are presented in this work. In all situations where we do not indicate explicitly how we computed the average synaptic activity both excitatory and inhibitory synapses are taken into account. RESULTS In this section all the values of spike rates, synaptic activities, external inputs and strengths of the connection are given in arbitrary units. The graphics corresponding to the effects of varying external inputs and strengths of the connections were obtained using twenty different values of the varying quantities. The graphics showing results in function of phase difference were obtained using forty different phase values. When the results are shown in function of time the resolution used is one millisecond. All the specific values of the parameters used in the results are chosen to illustrate an effect that is qualitatively equal for a range of parameters.

The results presented in this section were obtained for one brain area, modeled as illustrated in Fig. 2a. We wanted to investigate the relation between spike rates and synaptic activities when the external inputs were mainly to the excitatory neurons. For simplicity the input to the inhibitory neurons was taken to be zero (h i ⫽ 0). The results presented are steady-state solutions. First the behavior of the model was studied when the external input h e to the excitatory neuronal population was varied. Two cases are shown corresponding to two different and fixed pairs of values of the local connection strengths, namely, S e ⫽ 0.2 and S i ⫽ 0.5 (Figs. 3a and 3b) and S e ⫽ 0.8 and S i ⫽ 0.5 (Figs. 3c and 3d). Figures 3a and 3c show that the average spike rates and synaptic activities increase linearly with the external input. The average spike rate of the excitatory population and the synaptic activity increase proportionally. Their ratio is constant for varying external input and fixed values of strengths of the connections (Figs. 3b and 3d). However, the value of the ratio is different for the two situations. Figures 4a and 4b show how the relation between the spike rates and synaptic activity depends on the strength of the local excitatory connections (S e). The values of the strength of the inhibitory connections and external input considered are S i ⫽ 0.5 and h e ⫽ 1. In Fig. 4a it can be seen that the variations in spike rate of the excitatory population and synaptic activity with increasing S e are not proportional. Increasing intrinsic excitation S e (as in the presence of synaptic facilitation) induces a decrease in the ratio between the spiking activity of the excitatory population and synaptic activity (Fig. 4b, solid line). The solid line in Fig. 4b is obtained if both excitatory and inhibitory synapses are considered. The dashed line in Fig. 4b results if in the same setup only excitatory synapses are considered to contribute to the synaptic activity. The ratio decreases with increasing value of S e in both situations, although less pronouncedly for the case when only excitatory synapses are considered. The next question we addressed was how the system behaves when the strengths of the intrinsic inhibitory connections S i are varied. Figure 4c shows the dependence of the ratio between spiking and synaptic activity on S i for fixed external input h e ⫽ 1 and the value of the intrinsic excitatory connections S e ⫽ 0.5. The effect observed is analogous to the one resulting from variation in S e, but less pronounced. The effect of varying S i was studied considering the synaptic activity of excitatory and inhibitory synapses (solid line) and considering only excitatory synapses (dashed line). For the case where only excitatory synapses are taken into account, the ratio is almost constant.

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connectivity strength fixed at values S e ⫽ 0.1 and S i ⫽ 0.1 and constantly driving inhibitory interneurons with input h i ⫽ 0.5. When the external input to the excitatory population increases, the average spike rates and synaptic activity both increase (Fig. 5a). The spike rate of the inhibitory population has just a slight increase, when compared to the spike rate of the excitatory population and to the synaptic activity. The increases in spike rate of the excitatory population and synaptic activity are not proportional to each other. The ratio between these two quantities increases with h e both for the case when all synapses are considered (Fig. 5b, solid line) and when only excitatory synapses are taken into account (Fig. 5b, dashed line). The dependencies of the ratios on h e are very similar. An interesting effect appeared if we varied the strength of input to inhibitory interneurons, h i, while keeping the other quantities fixed at h e ⫽ 1, S e ⫽ 0.5, and S i ⫽ 0.5. In this situation we found that the average spike rate of the excitatory population and the

FIG. 3. (a) Behavior of the average spike rates (solid line for the excitatory population and dashed line for the inhibitory population) and of the synaptic activity (dash-dotted line) in function of the intensity of the external input to the excitatory population. (b) Ratio between the average spike rate of the excitatory population and the synaptic activity of the intensity in function of the external input to the excitatory population. The strengths of the connections are S e ⫽ 0.2 and S i ⫽ 0.5; (c) the same same as in (a) but for strengths of the connections equal to S e ⫽ 0.8 and S i ⫽ 0.5; (d) the same as in (b) but for strengths of the connections equal to S e ⫽ 0.8 and S i ⫽ 0.5.

Effect of Inhibition The results described in this section refer to the architecture represented in Fig. 2a and are solutions for the steady-state case. The difference from the setup used in the previous section is that now the external input to the inhibitory neuronal population is different than zero. With this setup we studied the effect of driving inhibitory interneurons on the relation between spiking and synaptic activity. We varied the strength of external input, h e, while keeping the local

FIG. 4. (a) Average spike rates and synaptic activity in function of the intensity of the strength of the excitatory connections. (b) Ratios between the average spike rate of the excitatory populations and the synaptic activities against the intensity of the strength of the excitatory connections, when synaptic activity takes into account only excitatory synapses (dashed line) or both excitatory and inhibitory synapses (solid line). (c) The same quantities as in (b) but of the intensity of the strength of the inhibitory connections.

SYNAPTIC ACTIVITY AND SPIKE RATES

FIG. 5. (a) Average spike rates and synaptic activity in function of the intensity of the external input to the excitatory population. (b) Ratios between the average spike rate of the excitatory populations and the synaptic activities against the external input to the excitatory population.

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FIG. 7. Behavior of the average spike rates and of the synaptic activity in time.

Effect of Neuronal Activation Thresholds synaptic activity have opposite behaviors (Fig. 6a). In such a system the total synaptic activity and the actual spiking activity of the system would be anticorrelated. The ratio between the spike rate of the excitatory population and synaptic activity decreases when h i increases, as shown in Fig. 6b. The results are similar whether we consider just excitatory synapses (dashed line) or inhibitory and excitatory synapses (solid line).

When the sum of the inputs that a excitatory neuronal population receives is not sufficient to induce the firing of action potentials, there is synaptic activity that does not reflect any spiking activity of the excitatory cells. This effect, which arises because neurons have an activation threshold, becomes prominent in our model under two conditions. The first condition is when the external input to the inhibitory population, h i, is sufficiently high to cause the summed input to the excitatory neuronal population to be below the firing threshold. This case is illustrated in Fig. 7, which shows the synaptic activity and spike rates in the area as a function of time. After a short transient the spike rate of the excitatory population is zero, while the synaptic activity is not. This is caused by the strong inhibition imposed on the excitatory neurons by their inhibitory companions. The values for the strengths of the connections and the external inputs are S e ⫽ S i ⫽ 1, h e ⫽ 1, and h i ⫽ 2. The second condition, in which neural thresholds become important, is the regime of weak external input. The system shows synaptic activity but no spiking activity, if the firing threshold of the excitatory population is higher than the input that the population receives. Effect of Synchronous Neuronal Firing

FIG. 6. (a) Average spike rates and synaptic activity in function of the intensity of the external input to the inhibitory population. (b) Ratios between the average spike rate of the excitatory populations and the synaptic activities against the external input to the inhibitory population.

In this section we address the way different states of synchrony can affect the relation between synaptic activity and spike rates. The results described are numerical solutions of a system of two connected brain areas, as illustrated in Fig. 1. An oscillatory external

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was fixed and equal to zero, while the initial phase of the input to area 1 was varied between zero and 2␲. For each phase difference the time averages of the spike rates and synaptic activity were calculated. The time averages were computed over 9830 ms, starting after the initial 170 ms. The first 170 ms were not taken into account, to avoid initial transient behaviors. The synaptic activities of area 1 (solid line) and area 2 (dashed line) are plotted in Fig. 9a against the initial phase of the external input to area 1. Figure 9b shows the relation between the average spike rates of the excitatory populations of areas 1 and 2 and the phase difference of the external inputs to the two areas. For both areas, there is a slight variation of both the synaptic activities (around 10%) and the spike rates (just a few percent) with the phase difference. The ratios between the spiking and synaptic activities vary only by about 10% (Fig. 9c). Also, the rates of the excitatory populations and the synaptic activities have different FIG. 8. (a) Time course of the external inputs to the excitatory population of area 1 (solid line) and of area 2 (dashed line). (b) Average spike rates and synaptic activity of area 1, as a function of time. (c) Average spike rates and synaptic activity of area 2, as a function of time.

input was imposed on the excitatory neuronal population of each brain area. This input can be thought to arise from two other parts of the brain, which are firing in synchrony already and thereby drive in an oscillatory way the two areas we considered. These other brain areas can fire in or out of phase, which causes the inputs to the model to have different phase relationships. Motivated by this scenario we investigated the influence different phases of synchronous firing can have on the ratio between synaptic metabolic activity and the mean spike rate. The inputs h e1 and h e2 to both areas are sinusoidal functions with a period of 340 ms, scaled and translated to oscillate between 0 and 1. The phase difference of the two inputs was varied. The exact frequency of oscillation considered is arbitrary and results qualitatively equal hold for other values of the frequency. Let us first consider the situation illustrated in Fig. 1, with values of the strengths of the connections equal to: S e1 ⫽ S e2 ⫽ S i1 ⫽ S i2 ⫽ L e1 ⫽ L e2 ⫽ 1 and L i1 ⫽ L i2 ⫽ 2. In Fig. 8a the inputs are plotted against time when the initial phase of the input to area 1 is ␲/2 and to area 2 is zero. For this situation, the behaviors of the average spike rates of the excitatory and inhibitory populations and of the average synaptic activity against time are shown in Fig. 8b for area 1 and in Fig. 8c for area 2. The figure demonstrates that the oscillatory external inputs induced oscillatory behavior of the spike rates and consequently of the synaptic activity. For the situation described above, the influence of the phase difference between the inputs on the system was studied. The initial phase of the input to area 2

FIG. 9. (a) Synaptic activity of area 1 (solid line) and of area 2 (dashed line) with regard to the phase difference between the external inputs. (b) Average spike rate of the excitatory neuronal population of area 1 (solid line) and of area 2 (dashed line) with regard to the phase difference between the inputs. (c) Ratio between the average spike rate of the excitatory neuronal population and the synaptic activity of area 1 and 2 against the phase difference between the inputs.

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depending on the relative phases of oscillatory activity in the brain, the average synaptic activity from a brain area can reflect either strongly spiking or completely silent excitatory neurons. DISCUSSION Model Setup and Robustness of Effects

FIG. 10. (a) Ratios between the average spike rate of the excitatory neuronal population and the synaptic activity of area 1 plotted against the phase difference between the inputs. (b) The same as in (a) but for area 2.

behaviors for the two areas. The same analysis considering that just the excitatory synapses contribute to the synaptic activity was performed. The results are summarized in Fig. 10. Figures 10a and 10b show the results for areas 1 and 2, respectively, as a function of phase difference. The behaviors of the ratios relative to areas 1 and 2 are similar (almost parallel) for both ways of computing the synaptic activities. We now consider the situation where there is just one unilateral connection between the brain areas, from the excitatory population of area 1 to the inhibitory population of area 2. This scenario is illustrated in Fig. 2b. The values of the strengths of the connections used are S e1 ⫽ S e2 ⫽ 1.2 and S i1 ⫽ S i2 ⫽ L i1 ⫽ 1. The average spike rate of the excitatory population and the average synaptic activity are plotted against the initial phase of the external input to area 1 in Fig. 11a. The average spike rate of the excitatory population and the average synaptic activity both decrease, then increase, and finally decrease again with the phase difference, reaching minimum values around ␲/6. The spiking and synaptic activities vary differently as the phase difference increases, leading to a change in the ratio between the two quantities. In fact, the ratio decreases all the way to zero for some intervals of phase differences. It then increases strongly and decreases again, as can be seen in Fig. 9b for the case in which both synapse types (solid) or only excitatory synapses (dashed) were considered. It becomes obvious that the ratio between the spike rate of the excitatory neuronal population and synaptic activity is strongly dependent on the state of synchronization between the oscillatory inputs received by the two brain areas. This result predicts that,

It is important to understand the relation between the signals measured by PET and fMRI and neuronal spiking activity. In particular it is crucial to understand the relation between PET and fMRI and the spiking activity of the excitatory neurons, because these cells are responsible for long-range cortical communication. PET and fMRI measurements reflect changes in local blood flow, which are thought to be associated with metabolic energy consumption, which in turn is believed to reflect mainly synaptic activity (for reviews see, e.g., Jueptner and Weiller, 1995, and Villringer, 1999). To discuss our work in the context of neuroimaging we from now on consider, based on the knowledge explicated in the Introduction, that imaging signals are more closely associated with synaptic than with spiking activity. Further, we consider that the relation between synaptic activity and imaging signal is mediated by local metabolic energy consumption. The energy consumption associated with a single postsynaptic potential can be considered proportional to its absolute integral. When many EPSPs and IPSPs are exchanged, within or between neuron pools, the corresponding synaptic energy consumptions can be

FIG. 11. (a) Average spike rate of the excitatory neuronal population and synaptic activity of area 2 with regard to the phase difference between the external inputs to areas 1 and 2. (b) Ratios between the average spike rate of the excitatory neuronal population and the synaptic activity of area 2 plotted against the phase difference between the inputs.

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assumed to sum up linearly. Hence, the total synaptic activity defined in our model can be taken to represent PET or fMRI signal measured in one area. The quantity relevant to imaging experiments corresponds to a temporal mean over a few seconds of the synaptic activity, Eqs. (7) and (8), as presented under Results. The aim of our work was to theoretically model the relation between synaptic activity and neuronal spike rates averaged over brain areas. Averages over brain areas were used to account for the limited spatial resolution of the PET and fMRI signals. The brain areas modeled were not associated with a particular cortical region and did not have a specific size. They represent general functional regions as detectable in a normal PET or fMRI experiment. Metabolic consumption is mainly associated with synaptic activity. Both excitatory and inhibitory synapses are thought to be involved (for reviews, see Jueptner and Weiller, 1995, and Villringer, 1999). However, Waldvogel and coauthors (Waldvogel et al., 2000) showed that inhibited regions in the brain might be associated with no change in the BOLD fMRI signal. They argued that even if the inhibitory synapses are metabolically active, they are less numerous and more efficient than the excitatory ones. This implies that the excitatory synapses could be the dominating factor in metabolic consumption, reflected at the scale of brain imaging techniques. Because this issue is still under discussion, we took into account two possible extremes for the source of activity-related energy uptake: One considers that the relevant synaptic activity is by equal parts from excitatory and inhibitory synapses, the other considers only the synaptic activity of the excitatory synapses. We found that all our results depended only weakly on the case chosen; they proved to be robust against the unknown relative contribution of inhibitory synapses. For all the situations presented the shape of the PSP [Fig. 13c and Eq. (11)] was kept constant and equal for all neuronal populations, both excitatory and inhibitory. This choice was made for the sake of simplicity and because we were modeling a general brain area with general excitatory and inhibitory cells. We did not commit ourselves to a particular type of cell or cortical region. However, using different values for the parameter ␶ a, a ⫽ e, i, associated with each neuronal population, does not qualitatively change our results: Actually the function used for the PSP can influence only the non-steady-state solutions; the steady-state results are independent of the PSP shape. For the non-steadystate solutions, however, the curve shape of the PSPs only affects the quantitative value of the oscillation frequency, from which we did not draw any conclusions or predictions. Also, the actual values and ranges of values considered for the strengths of the connections and the external inputs are just examples of possible values lead-

ing to certain effects. The same types of effects as the ones described under Results were still observed for other values of the parameters, around the ones chosen. Interpretation of PET and fMRI Signals We studied a system of one or two connected brain areas receiving external inputs. Different situations are presented, where the local connectivity, the connectivity between areas, the external inputs and the dynamical state of the system varied. In these situations, mismatches between synaptic and spiking activity were found, which can pose a problem to the interpretation of neuroimaging data. Such mismatches make it difficult to integrate results of experiments using techniques which measure synaptic and spiking activities, for instance, integrating fMRI and PET signals with multi- or single-unit electrophysiological recordings. It also makes it difficult to infer functional connectivity between brain areas, based on their activities as measured by PET or fMRI, as in studies using structural equation modeling (McIntosh et al., 1994) or correlation analysis (Horwitz et al., 1984). The difficulty is in the fact that different areas can only influence each other by excitatory spiking activity. Conclusions about connectivity based on signals reflecting synaptic activity can be erroneous, once relative differences in synaptic activity do not necessarily reflect the differences in spiking activity. There might even be increases in synaptic activity, and so in imaging signal, that are associated with decreases in spiking activity and synaptic activity which is not associated with spiking activity at all. Below each of the analyzed situations is discussed in detail, following the organization of the results section. In the first subsection of the Results a system of one brain area receiving external input only to the excitatory population was analyzed. The ratio between the spike rate of the excitatory population and synaptic activity is constant, as a function of the external input to the excitatory population (Fig. 3). In this case an increase in measured PET or fMRI signal reflects the same increase in underlying average excitatory spiking activity. Electrophysiology results, averaged in both time and space, should be reflected in equivalent imaging experiments, except for a multiplicative factor. In fact, this relation between average electrophysiology and neuroimaging signals has been inferred experimentally. Rees and coauthors (Rees et al., 2000) and Heeger and coauthors (Heeger et al., 2000) showed empirically a proportional relation between average spike rates measured in monkeys and fMRI signal measured in humans in V5 and V1, respectively. In these works fMRI and firing activity were shown to increase proportionally with the coherence of moving visual stimuli in V5 and with the visual stimulus con-

SYNAPTIC ACTIVITY AND SPIKE RATES

trast in V1. However, our simulations indicated that the proportionality factor between the spiking and synaptic activities depends on the local connectivity. The results presented show that the ratio between the two quantities decreases with increases in the strengths of the local excitatory or inhibitory connections (see Fig. 4). A strong dependence of the proportionality factor on the particular experiments is also reported in the literature (Heeger et al., 2000). The dependency can originate from differences in the experimental setup but also from the effect described above. Our model predicts that under the experimental conditions investigated by Rees and Heeger there is no significant external input to the inhibitory neurons and no significant change in local connectivity with the stimuli variations. Generally speaking, the variability of the proportionality factor between spiking and synaptic activity implies that quantitative comparisons between values of imaging experiments can be erroneous. A difference detected in PET and fMRI signals does not necessarily reflect a proportional difference in excitatory spiking activity. There are two possibly problematic situations. The first is when the different imaging signals originate in two brain areas with different local connectivity. The second is when the signals are acquired under different experimental conditions associated with changes in the strength of local connectivity. The last situation can occur as a result of synaptic adaptation or learning and is probably common in many experimental setups. Recently a study was published directly comparing electrophysiology with fMRI measurements (Logothetis et al., 2001). One of the results shown was a mismatch between multiunit electrophysiological measurements and the fMRI BOLD signal, in the presence of constant stimulation. It was shown that for a fraction of neurons the spiking activity died out completely after a few seconds, whereas the fMRI response persisted for all the stimulus presentation and for some interval after (see Fig. 3a of Logothetis et al., 2001). This observation could reflect the effect of modifications of synaptic strengths, as modeled in the first subsection under Results and discussed above. We illustrate this effect by providing a simulation, which reproduces the experimental findings mentioned above, given that we can identify synaptic activity of the model with the amplitude of measured local field potentials and the spiking activity of the model with measured multiunit activity. Synaptic plasticity is modeled by making the strengths of the local connectivities of one brain area dependent on time. The excitatory local connections decrease with time, while the inhibitory ones increase. The time dependency of the local connections is taken to be S e共t兲 ⫽ 0.7 ⫺ 0.3共1 ⫺ e ⫺t/1000兲 ,

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FIG. 12. (a) Average spike rate of the excitatory population. (b) Average synaptic activity.

S i共t兲 ⫽ 0.3 ⫹ 0.7共1 ⫺ e ⫺t/1000兲 . The external inputs to the excitatory and inhibitory neuronal populations used were h e ⫽ 1 and h i ⫽ 2. Figure 12 shows the behavior of the average spike rate of the excitatory population and the average synaptic activity, plotted against time. The figure shows that, in agreement with the experimental finding, there is a transient spiking activity going to zero at about 2500 ms, while the synaptic activity converges to a constant value different from zero. Our results provide one possible explanation of the experimental observations, but it should be noted that there might be several other factors that could explain the results by Logothetis and coauthors. The effect of varying the external input to the inhibitory population, for fixed values of the local connectivity and external input to the excitatory population, is particularly striking (Fig. 6). In this case the spike rate and synaptic activity have opposite behaviors. To illustrate the implications consider, for example, two connections inferred using structural equation modeling or correlations, from one area A to two areas, B and C. Suppose that areas B and C receive different external input to their inhibitory neurons, all the other parameters being equal. In this situation the synaptic activities and spike rates for the two areas can be anticorrelated (see Fig. 6). Hence, the connection corresponding to lower firing rates and less communication between areas could be detected as the strongest. In the third section under Results we examined situations where the summed input received by the excitatory population was below the neuronal activation threshold. In this situation there is synaptic activity and so PET and fMRI signal that corresponded to no

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excitatory spiking activity. In imaging experiments, areas have been detected in association with attentional effects (see for example, the work of Kastner et al., 1998). These areas of increased synaptic activity might reflect silent neurons, receiving only subthreshold facilitatory input, which causes the attention effect. This is an example where one should be cautious in the interpretation and modeling of the results. Finally, in the last subsection under Results we investigated the effect of synchronization of imposed oscillatory behavior of two brain areas. The idea is that two connected brain areas with oscillatory average spiking behavior will tend to boost or suppress each others’ activities, depending on the state of synchronization of their oscillations. A system of two connected brain areas as we modeled them can go into an oscillatory regime with varying synchronization, even for constant external inputs (not shown), but here we considered oscillatory external inputs, to lock the system to a regular behavior associated with a given phase difference, greatly simplifying the analysis of the results. The oscillatory external inputs can be thought to correspond to inputs from other brain areas having oscillatory behavior. Such dynamic states are believed to occur in the brain (Singer, 1999). The effect of synchronization was studied for two recurrently connected brain areas (see Fig. 1). The ratios, for the two areas, between spike rate of the excitatory population and synaptic activity depend on the synchronization of the input, but the effect is relatively small. The ratios plotted against phase difference have opposite behaviors for the two areas. This means that if such a situation occurs in an imaging experiment, increases in PET or fMRI signal in two brain areas could correspond to a decrease in spike rate in one area and to an increase in the other area. If one brain area drives the other by feed-forward inhibitory connections (Fig. 2b), the relation between the spike rate of the excitatory population and synaptic activity, in the driven area, is strongly dependent on the synchronization of the oscillations of the two areas. Neuronal synchronization has been proposed to play an important role in cognition (for a review see Singer, 1999). Following this view, different states of synchronization might be associated with different cognitive states. If this is the case, according to our theoretical model, similar neuroimaging signals acquired in such different cognitive states might be associated with very different spike rates. The decoupling between spiking and synaptic activities observed for the feed-forward case is stronger than the one observed for the recurrently connected case. This is intuitively expected since in the recurrently connected case the spiking behavior of each area is more closely related to its input (and so to its synaptic activity). In fact, the spiking activity of one area indirectly determines part of its own input, via the

other area. A weak decoupling for the recurrently connected case might generalize for other effects apart from that of synchronization. If this is the case and if recurrently connected architectures are predominant in the brain, we predict neuroimaging data to be relatively reliable. Related Work Attempts at relating neuronal spiking activity and functional imaging signals have been presented previously by Arbib et al. (1995), Scannell and Young (1999), Rees et al. (2000), Heeger et al. (2000), Logothetis et al. (2001), and Tagamets and Horwitz (1998 and 2001). In this section we discuss the work of Tagamets (Tagamets and Horwitz, 2001) since it is closely related to our own work. Both works address the relation between spiking and synaptic activities using mean-field formulations, although applied to different models of brain areas. Tagamets and Horwitz used the meanfield formulation of Wilson and Cowan (1972), while we used a modified version of the formulation by BenYishai et al. (1995). For the cases of external inhibition and synaptic plasticity, our work confirms, for a different model, the results previously reported. It adds comparisons with experimental works reported by Rees et al. (2000), Heeger et al. (2000), and Logothetis et al. (2001). It also adds results referring to the effect of neuronal thresholds and the effect of synchronization of oscillatory behavior. Finally, we also analyzed the effect of considering that both excitatory and inhibitory synapses or just excitatory synapses contribute to metabolic consumption. APPENDIX 1 Here we provide the mathematical formulation of a mean-field pool, its input, and its spiking activity, which provide Eqs. (1)–(5). Figure 13a shows a set of N e excitatory neurons, which receive synaptic connections from each other as well as some external input. If we denote the set of spike firing times for neuron l in the pool by {t lf}, the total synaptic input received by neuron k at time t is given by

冘 冘 E 共t ⫺ t 兲 ⫹ h Ne

H e,k共t兲 ⫽

kl

l⫽1 t f 再 l冎

冘冘S

f l

e,k共t兲

,

(12)

N

⫽

l⫽1 t f 再 l冎

e,kl␧ e共t

⫺ t lf 兲 ⫹ h e,k共t兲 ,

(13)

where S e,kl is the excitatory synaptic weight from neuron l to neuron k, E kl(t) is the EPSP evoked in k by a spike of l, and h e,k(t) summarizes the total input to k

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The synaptic input seen at present contains a contribution n e(t 0)S e␧ e(t ⫺ t 0) which originates from the n e(t 0) spikes fired at time t 0 in the past (Fig. 13c). H e is the same for all neurons in the pool—it does not depend on k anymore—and describes the mean synaptic input of the population. Now we introduce the instantaneous mean spike rate m e(t) as the fraction of neurons firing a spike at t, m e(t) ⫽ n e(t)/N e. Using this definition and taking the continuous limit in Eq. (15), the mean synaptic input becomes

H e共t兲 ⫽

冕

⬁

S e␧e共t⬘兲me共t ⫺ t⬘兲dt⬘ ⫹ he共t兲 .

(16)

0

Similarly, the mean synaptic input for a pool of inhibitory neurons is given by

H i共t兲 ⫽ ⫺

冕

⬁

S i␧i共t⬘兲m i共t ⫺ t⬘兲dt⬘ ⫹ hi共t兲 .

(17)

0

FIG. 13. (a) A set of N neurons mutually connected with identical connection strengths and receiving identical external input, h 0, from a pool. (b) The mean activity is the fraction of active neurons at a small time slot after t 0. (c) The summed postsynaptic potential generated by the active neurons at t 0.

from outside the pool. Now we can rearrange the sums in Eq. (12) to combine the spike firing times of any of the N e neurons in the pool, which occur within small time slots, say between t 0 and t 0 ⫹ dt (Fig. 13b). This yields H e,k共t兲 ⫽

冘 冘

t0 t f 兩t ⱕt fⱕt ⫹dt 再 l 0 l 0 冎

S e,kl␧ e共t ⫺ t lf 兲 ⫹ h e,k共t兲 .

(14)

The first term on the righthand side of Eq. (16) corresponds to Eq. (1) under Methods. If a neuron pool receives input not only from itself but also from other neuron populations, the inputs from all presynaptic populations sum to yield the total synaptic input. For the architecture with four pools (Fig. 1), the resulting inputs are given by Eqs. (2)–(5). APPENDIX 2 Expression (6) for the dynamics of the mean spike rate (Stetter, 2001) is derived by modifying existing approaches (Ben-Yishai et al., 1995; Gerstner, 2000). We start by considering individual neurons to be binary stochastic units, which can flip forth and back between an active (spiking) and inactive (nonspiking) state. The probability per unit time for an inactive neuron to be activated is denoted by the activation rate ␥; the rate by which the neuron becomes inactivated is denoted by ␦. If we consider a population of N binary stochastic neurons with identical activation and inactivation rates, the number, A, of active neurons changes within the small time interval ⌬t according to

By use of the mean-field assumption we can replace the individual synaptic efficacies S e,kl by their mean value, S e/N e. This is equivalent to assuming that the neurons within the pool considered are uniformly connected to each other. The denominator N e assures that the total input to a neuron is independent of the system size. Further, we can replace the actual external input h e,k by its mean value, h e. As the time slots considered are small compared to typical interspike intervals, we ⌬A ⫽ ␥ ⌬t共N ⫺ A兲 ⫺ ␦ ⌬tA ⫽ 共 ␥ N ⫺ 共 ␥ ⫹ ␦ 兲A兲⌬t . (18) also can approximate the actual spike firing times within each slot by the onset time t 0 of the slot. If the In the limit ⌬t 3 0, this relation is transformed into slot at t 0 contains n e(t 0)dt spikes (Fig. 13b), Eq. (14) a rate equation for the fraction of neurons m ⫽ A / N becomes that are active at time t: H e共t兲 ⫽

冘 nN共t 兲 S ␧ 共t ⫺ t 兲dt ⫹ h 共t兲 . e

0

e e

t0

e

0

e

(15)

d dt

m ⫽ ␥ ⫺ 共 ␥ ⫹ ␦ 兲m .

(19)

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Now we assume for simplicity that the inactivation rate behaves inversely to the activation rate. This is reasonable because neurons which are strongly driven by input are less likely to stop firing spontaneously. If ␥ max denotes the maximum possible activation rate, we arrive at ␦ ⫽ ␥ max ⫺ ␥ and

late the dependence of the activation probability g on the mean synaptic input H of the population as g共H兲 ⫽ max共␤共H ⫺ T兲, 0兲 .

(22)

ACKNOWLEDGMENTS d dt

m ⫽ ␥ ⫺ ␥ maxm.

(20)

One possible interpretation of the maximum activation is related to the refractory period of biological neurons. For a neuron to undergo two subsequent activations, it must at least fire a spike and wait for the absolute refractory period ␶ until it can be activated to fire the next spike. Thus it seems reasonable to identify the maximum activation rate approximately with ␥ max ⫽ 1/␶. The rate equation for the population activity becomes

␶

d dt

m ⫽⫺m ⫹ ␶␥ ⫽: ⫺m ⫹ g,

(21)

where 0 ⱕ g ⫽ ␶␥ ⱕ 1 denotes the activation probability (relative to ␶) for the neuron population. In combination with Eq. (22) below, this corresponds to Eq. (6) under Methods. For realistic regimes of operation, Eq. (21) can be interpreted as the dynamics of a pool of spiking neurons: The (absolute) refractory period ␶ is in the range of 1–2 ms, which means that electrically driven neurons can reach spike rates of approximately 500 –1000 Hz. In contrast, the spike rates observed for naturally driven cortical neurons range around 50 Hz: realistic activation rates are much smaller than the maximal rate, ␥ Ⰶ ␥ max, (g Ⰶ 1), and consequently the inactivation rate is very fast: ␦ ⬇ ␥ max ⫽ 1/␶. In this regime, each time a neuron becomes activated and fires a spike, it immediately becomes inactivated again with a rate close to its refractory period. In other words, a binary stochastic neuron which is activated fires an individual spike and automatically inactivates again. Consequently, in the limit of low mean activation, the rate equations (6) and (21) describe the time evolution of the mean activity of a population of spiking (instead of binary) neurons. Finally we must specify, how the activation probability changes with the synaptic input. If the input of a neuron population falls below a threshold T, all neurons of the population are inactive or just spontaneously active. Otherwise neurons should be activated more frequently the more excitatory input they receive. The simplest function, which preserves this important rectifying nonlinearity present in biological neuronal systems, is a semilinear function, and we can formu-

This work was supported by the EU Advanced Course in Computational Neuroscience (EU Grant HPCFCT-1999-00232, IBRO Neuroscience Schools and Boehringer Ingelheim) and the Foundation for Science and Technology, Portuguese Ministery for Science and Technology (PRAXIS XX1/BD/13979/97). We thank Jonas Larsson, Anders Ledberg, and Per Roland for helpful comments and suggestions.

REFERENCES Arbib, M.-A., Bischoff, A., Fag, A. H., and Grafton, S. T. 1995. Synthetic PET: Analyzing large-scale properties of neural networks. Hum. Brain Mapp. 2: 225–233. Bandettini, P. A., and Ungerleider, L. G. 2001. From neuron to BOLD: New connections. Nature Neurosci. 4(9): 864 – 866. Ben-Yishai, R., Bar-Or, R. L., and Sompolinsky, H. 1995. Theory of orientation tuning in visual cortex. Proc. Natl. Acad. Sci. USA 92(9): 3844 –3848. Gerstner, W. 2000. Population dynamics of spiking neurons: Fast transients, asynchronous states and locking. Neural Comput. 12(1): 43– 89. Heeger, D. J., Huk, A. C., Geisler, W. S., and Albrecht, D. G. 2000. Spikes versus BOLD: What does neuroimaging tell us about neuronal activity? Nature Neurosci. 3(9): 631– 633. Heeger, D. J., and Ress, D. 2002. What does fMRI tell us about neuronal activity? Nature Rev. Neurosci. 3(2): 142–151. Horwitz, B., Duara, R., Rapoport, S. I. 1984. Intercorrelations of glucose metabolic rates between brain regions: Application to healthy males in a state of reduced sensory input. J. Cereb. Blood Flow Metab. 4(4): 484 – 499. Jueptner, M., Weiller, C. 1995. Review: Does measurement of regional cerebral blood flow reflect synaptic activity?—Implications for PET and fMRI. NeuroImage 2(2): 148 –156. Kadekaro, M., Crane, A. M., Sokoloff, L. 1985. Differential effects of electrical stimulation of sciatic nerve on metabolic activity in spinal cord and and dorsal root ganglion in the rat. Proc. Natl. Acad. Sci. USA 82(17): 6010 – 6013. Kastner, S., Weerd, P. D., Desimone, R., and Ungerleider, L. G. 1998. Mechanisms of directed attention in the human extrastriate cortex as revealed by functional MRI. Science 282(5386): 108 –111. Logothetis, N. K., Pauls, J., Augath, M., Trinath, T., and Oeltermann, A. 2001. Neurophysiological investigation of the basis of the fMRI signal. Nature 412(6843): 150 –157. Magistretti, P. J., and Pellerin, L. 1996. Cellular bases of brain energy metabolism and their relevance to functional brain imaging: Evidence for a prominent role of astrocytes. Cereb. Cortex 6(1): 50 – 61. Magistretti, P. J., and Pellerin, L. 1999. Cellular mechanisms of brain energy metabolism and their relevance to functional brain imaging. Philos. Trans. R. Soc. London Ser. B—Biol. Sci. 354(1387): 1155–1163. McIntosh, A. R., Grady, C. L., Ungerleider, L. G., Haxby, J. V., Rapoport, S. I., and Horwitz, B. 1994. Network analysis of cortical visual pathways mapped with PET. J. Neurosci. 14(2): 655– 666.

SYNAPTIC ACTIVITY AND SPIKE RATES Nudo, R. J., and Masterton, R. B. 1986. Stimulation-induced [14C]2deoxyglucose labeling of synaptic activity in the central auditory system. J. Comp. Neurol. 245(4): 553–565. Rees, G., Friston, K., and Koch, C. 2000. A direct quantitative relationship between the functional properties of human and macaque V5. Nature Neurosci. 3(7): 716 –723. Scannell, J. W., and Young, M. P. 1999. Neuronal population activity and functional imaging. Philos. Trans. R. Soc. London Ser. B—Biol. Sci. 266(1422): 875– 881. Singer, W. 1999. Neural synchrony: A versatile code for the definition of relations? Neuron 24(1): 49 – 65. Stetter, M. 2001. Exploration of Cortical Function. Kluwer Scientific, Dordrecht. Tagamets, M.-A., and Horwitz, B. 1998. Neuronal population activity and functional imaging. Cereb. Cortex 8(4): 310 –320. Tagamets, M.-A., and Horwitz, B. 2001. Interpreting PET and fMRI measures of functional neural activity: The effects of synaptic

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inhibition on cortical activation in human imaging studies. Brain Res. Bull. 54(3): 267–273. Villringer, A. 1999. Physiological changes during brain activation. In Functional MRI (C. Moonen and P. Bandettini, Eds.), pp. 3–13. Springer Verlag, Berlin. Villringer, A., and Dirnagl, U. 1995. Coupling of brain activity and cerebral blood flow: Basis of functional neuroimaging. Cerebrovascular and Brain Metab. Rev. 7(3): 240 –276. Waldvogel, D., van Gelderen, P., Muellbacher, W., Ziemann, U., Immisch, I., and Hallett, M. 2000. The relative metabolic demand of inhibition and excitation. Nature 406(6799): 995–998. Wilson, H. R., and Cowan, J. D. 1972. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12(1): 1–24. Wilson, H. R., and Cowan, J. D. 1973. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13(2): 55– 80.