2.1 Relationship between LFP and BOLD: Analytic framework

The fMRI BOLD signal and the LFP measure neuronal population activity in a fundamentally different manner. The goal of this analytic framework is to capture these differences in simple mathematical expressions and from these expressions, derive the relationship between the two instrument measurements. We purposely omit a large number of biophysical details, such as cell types, neuronal compartments, the dynamics of blood flow, and so forth, both for tractability and in order to emphasize the basic principles of how different measures integrate neuronal activity. In the sections that follow, we then show that, when coupled to simulated neural responses, the model can account for many important patterns observed in fMRI and ECoG data from human visual cortex.

For this analytic framework, we consider how a population of n neurons responds to a stimulus or task during a brief epoch (time 0 to T), assumed to be on the order of a second. Each neuron will produce a time-varying dendritic current, denoted as I_i(t) for the ith neuron, resulting from the trans-membrane potential. We would like to know how these currents, I(t), relate to the fMRI BOLD signal and to the LFP signal measured by an ECoG electrode.

We assume that the LFP arises primarily from dendritic membrane currents [2]. We ignore output spikes. (Although spikes can influence the LFP [30], it is generally thought that the influence is smaller than synaptic and dendritic currents [2], and including spikes would not change the logic of our arguments.) For the ith neuron, the contribution to the LFP is then α_i × I_i(t). The constant α_i depends on the distance and orientation of the neuron with respect to the electrode, as well as the electrode’s impedance. For simplicity, we assume that each neuron is equidistant from the electrode and has the same orientation, like pyramidal neurons perpendicular to the cortical surface, and therefore its contribution to the electrode measurement is scaled by the same constant, α. These neurons act together like a single, equivalent circuit, and hence the LFP time series will sum the contribution from each neuron, (Eq 1)

Field potential recordings are often summarized as the power (or band-limited power) in the time series [31]. Here we summarize the LFP response within a short time window as the power in the signal summed over the time window T: (Eq 2)

Importantly, Eq 2 is a linear/nonlinear (L/N) computation, since the LFP power is computed by first summing the signals (L), and then computing the power (N).

The BOLD signal pools neural activity in a fundamentally different manner because it depends on metabolic demand [e.g., for reviews, see 1, 32]. (Recent work on the neurobiology of neurovascular coupling indicates that much of the BOLD signal is not caused directly by changes in the level of metabolic products such as glucose, but rather by signaling molecules that tend to correlate with metabolic demand [33]). For simplicity, we discuss the BOLD signal throughout in terms of metabolic demand and return to this issue explicitly in the Discussion, (3.6 The role of a simple model in understanding the relation between BOLD and LFP). The metabolic demand of each neuron will increase if the cell depolarizes (excitation) or hyperpolarizes (inhibition) [28, 34, 35]. Hence, the metabolic demand of a neuron is a nonlinear function of its membrane potential: either a positive or negative change in voltage relative to resting potential causes a current, thereby resulting in a positive metabolic demand. There are many possible nonlinear functions one could assume to summarize the metabolic demand from the dendritic time series, such as the rectified signal (absolute value) or the power (squared signal). For tractability, we assume the metabolic demand of the ith neuron is proportional to the power in the time-varying trans-membrane current integrated over time: β_i × (POWER(I_i(t)) or , with β_i a scaling constant for the ith neuron. (Similar results were obtained if we used the absolute value rather than the power). For the entire population of n neurons, we then assume the BOLD signal will sum the metabolic demand of each neuron. For simplicity, we use the same scaling constant for each neuron: (Eq 3)

Importantly, Eq 3 is an N/L computation, since the power is computed first (N) and then the signals are summed (L), opposite to the order of operations for the LFP in Eq 3 (Fig 1) (personal communication from David J Heeger). In other words, we approximate the BOLD signal as the sum of the power, and LFP as the power of the sum, of the separate neuronal time series. The difference in the order of operations can have a profound effect on the predicted signals, as in the simple example with two neurons depicted in Fig 1C and 1D. The BOLD signal pooled over the two neurons is the same whether the time series from the two neurons are in phase or out of phase, whereas the LFP power is large when the time series are in phase and small when they are out of phase.

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Fig 1. Pooling with different orders of operations can have a large effect on measured brain signals.

(A) The approach to directly correlate local field potential (LFP) and blood-oxygen-level dependent (BOLD) data. (B) The current approach to relate the LFP and BOLD from the same neuronal population activity. (C) In this illustration, the currents of two neurons (x₁ and x₂) has the shape of a sinusoid with noise, and the sinusoid is in phase between the two neurons. In the simulated electrode measurement, the signals are summed and the power is calculated (POWER[SUM] = 2.00). In the simulated measurement of metabolic demand, the power of each of these neurons is first calculated and then summed across the neurons (SUM[POWER] = 1.01). Here, the LFP and BOLD are both large. (D) In this illustration, the membrane currents of two neurons (x₁ and x₂) are the same as in panel (C) except that the two time series are in counterphase. Here, unlike (C), the LFP is nearly 0 and the BOLD signal is large (code to reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script01_Fig1.m).

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These approximations allow us to make predictions about the relation between LFP and BOLD. By theorem, we know that the power of the sum of several time series is exactly equal to the sum of the power of each time series plus the sum of the cross-power between the different time series (Eq 4): (Eq 4)

Applying this theorem to Eqs 2 and 3 shows the relationship between our models of BOLD and LFP power: (Eq 5)

We can now see that the LFP power depends on two quantities, one of which is related to the BOLD signal, and one of which is unrelated to the BOLD signal (Eq 5). The first quantity summarizes the total level of neural activity (summed across neurons), and the second quantity summarizes the relationship between neural time series (the cross-power, similar to covariance). If and when the second term tends to be large compared to the first, then the LFP power will not be closely related to the BOLD signal.

One cannot deduce from first principles whether the first term in Eq 4 (summed power) or the second term (summed cross-power) will dominate. However, the number of elements contributing to the two terms is quite different: For n neurons, the first term has n numbers (the power in each neuron’s time series), whereas the second term has nearly n² numbers (all the pairwise cross-powers). Hence, if there is any appreciable covariance, then the LFP power will be dominated by the second term, and the correlation with BOLD will be weak (except in cases where the cross-power and power are highly correlated).

To see how these equations translate to quantitative measures of BOLD and LFP, we consider a small neuronal population whose time series conform to a multivariate Gaussian distribution. We assume that each neuron’s time series has the same mean, m; the same variance, σ²; and all of the pairwise correlations have the same value, ρ: (Eq 6)

X is the population time series, μ is the mean of each time series, and ∑ is the covariance matrix. We can now rewrite the simulated BOLD signal (the sum of the power) and the LFP (power of the sum) in terms of the parameters of the multivariate Gaussian (and arbitrary scaling factors α, β), (Eq 7) where n is the number of neurons. This enables us to visualize how the BOLD signal and the LFP power depend on just three values: the variance, correlation, and mean in the neural time series, rather than on all the individual time series (Fig 2). For these neuronal time series, the LFP, modeled as the power of the sum of neuronal time series (panel A), is dominated by the neuronal cross-power (panel C). The BOLD signal, modeled as the sum of the power in the neuronal time series (panel B), makes little contribution to the LFP, except when the correlation between neurons is low (ρ is close to 0); in this case, there is no cross-power, and BOLD and LFP power are correlated.

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Fig 2. The influence of time series parameters on the power of the sum, the sum of the power, and the cross-power.

(A) LFP power, computed as the power of the sum of 5 time series from a multivariate Gaussian distribution (Eq 6). The LFP power is shown as a function of the correlation (ρ), variance (σ), and mean (μ) of the time series (Eq 7). (B) The same as A, except plotting the sum of the power rather than the power of the sum, in order to model the BOLD signal. (C) The same as B but for cross-power. The power of the sum—Panel A—is the sum of the terms in Panels B & C. (Code to reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script02_Fig2.m).

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2.2 Simulating the LFP and BOLD responses

In section 2.1, we proposed formulae to derive instrument measures from neuronal population activity. Here, we ask how we might simulate neuronal activity with a small number of parameters. A low-dimensional characterization of the population activity is useful since we normally do not have access to the time series of an entire population of neurons. Moreover, a low-dimensional representation can lead to better understanding and generalization even when high-dimensional data are available [36, 37]. After simulating the population activity, we then use the analytic framework from section 2.1 to compute the BOLD and LFP signals. The parameters for the simulations were fit to ECoG recordings from human V1, V2, and V3 [38]. Because there were recordings from multiple electrodes and multiple stimuli, we ran multiple simulations fit to the different ECoG responses. We then used these simulations to predict the BOLD signal and compared these predictions to the measured BOLD signal for the same stimuli and same cortical locations (but in different observers). The steps for simulating the neuronal population data and the derived LFP and BOLD, and for comparing the simulations to empirical data, are summarized in Table 1.

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Table 1. A summary of analysis steps for simulations (MODEL) and comparison to data (DATA).

https://doi.org/10.1371/journal.pbio.2001461.t001

In the ECoG experiments, there were four grating stimuli of different spatial frequencies, three noise patterns with different power spectra, and one blank stimulus (mean luminance). For each of the 8 stimuli and each of 22 electrodes in V1, V2 and V3, we decomposed the measured ECoG responses into three spectral components: broadband, narrowband gamma, and alpha (Fig 3). An important feature of this dataset is that the three components of the ECoG responses showed different patterns across stimuli [38]: stimuli comprised of noise patterns caused large broadband increases but little to no measureable narrowband gamma response, whereas grating stimuli elicited both broadband increases and narrowband gamma increases. Gratings and noise stimuli both resulted in decreases in alpha power compared to baseline (also see S1 Fig). Had all three responses been tightly correlated with each other, it would not be possible to infer how each relates separately to the BOLD signal.

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Fig 3. Decomposing electrocorticographic (ECoG) data into three summary components.

(A) A schematic to show the summary metrics derived from ECoG spectra: broadband power elevation (bb), narrowband gamma (ϒ), and alpha (α). Broadband was calculated by the increase in a 1/fⁿ signal, gamma was calculated by fitting a Gaussian on top of the 1/fⁿ line, and alpha was calculated as the difference from baseline in the alpha-frequency range. (B) Power spectrum for one example electrode during a blank stimulus (black), gratings (red), and noise patterns (blue). (C) From the power spectrum, changes in broadband, gamma, and alpha were calculated. These values were bootstrapped 100 times across trials. Error bars represent 68% confidence intervals. (code to download data and reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script03_Fig3.m).

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2.2.1 Simulations of BOLD and LFP responses from neuronal population activity.

Cortical neurons receive a large number of inputs from diverse cell types. For our low-dimensional parameterization of the population activity, we assumed that each neuron received a mixture of three types of inputs (Fig 4A). These three inputs, following summation and leaky integration, produce the three spectral components observed in the ECoG data. Input 1 approximated Poisson-like spike arrivals (C¹, “Broadband”) and had a mean above 0 (excitatory). Input 2 was a high-frequency oscillation, peaked between 40 Hz and 60 Hz, coordinated between neurons (C², “Gamma”), with a mean of 0. Input 3 was a low-frequency signal peaked between 8 Hz and 12 Hz that was inhibitory: i.e., the mean was below 0 (C³, “Alpha”).

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Fig 4. Simulated local field potential (LFP) and blood-oxygen-level dependent (BOLD).

(A) Three different inputs to each neuron were simulated: a broadband, random input with a small positive offset (C¹), an oscillatory input with a time scale of 40 Hz to 60 Hz (C²), and a negative input with a time scale of 10 Hz (C³). (B) The 3 inputs (C¹, C², C³) were summed in each neuron to produce the total input to the neuron. (C) The total input was passed through a leaky integrator to produce the dendritic dipole current (I_i). The LFP was simulated by summing the dendritic currents. (D) The BOLD signal was simulated by taking the power of the dendritic current for each neuron and then summing across neurons. (Code to simulate data reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script07D_Fig4.m).

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For each simulated neuron i, the total input C on each trial is the sum of these three signals (Fig 4B): (Eq 8)

We then passed the summed input in each neuron through a leaky integrator to produce the time-varying dendritic current for that neuron (I_i, Fig 4C): (Eq 9)

The membrane time constant τ reflects the time dependence of the trans-membrane current [39, 40]. In total, we modeled a population of 200 neurons, each of which produced a 1-second time series on each trial. From the neuronal population simulations, we computed the LFP and BOLD signals according to the equations above (section 2.1 “Relationship between LFP and BOLD: Analytic framework”). In brief, the LFP was computed by summing the trans-membrane current across neurons (Eq 1, Fig 4C), and the BOLD signal was computed by summing the power across neurons (Eq 3, Fig 4D). The LFP was used as training data (to fit the parameters of the inputs), and the BOLD signal was used as test data (to test the accuracy of the model).

2.2.2 Simulation inputs produce three effects in the LFP power spectrum.

Below, we explain how the time series was generated for each of the three types of inputs and what kind of effect a change in each input has on the power spectrum.

Broadband input (C¹). Input C¹ was Gaussian white noise with a small positive bias. The Gaussian white noise approximates Poisson-distributed spike arrivals, each of which produces a small positive or negative conductance change, corresponding to excitatory or inhibitory postsynaptic potentials. The small positive bias reflects the assumption of more excitatory than inhibitory synaptic currents, causing a net depolarization. Gaussian white noise was used rather than Poisson-distributed synaptic inputs for computational efficiency, but the pattern of results is similar for Poisson or Gaussian distributions. For the purposes of simulations, we defined a high value of C¹ as high variance in the Gaussian distribution and low values of C¹ as low variance. This mimics the effect of high versus low rates of spike arrivals. The time series for the 200 neurons was generated from a distribution with 0 correlation for all pairs of neurons. Because the variance differed across simulations, and the correlations were always 0, the possible C¹ values span a vertical slice of the plots in Fig 2 (ρ-axis = 0). This white noise input, after passing through leaky integration, results in an output whose power spectral density declines with temporal frequency. When this input increases (higher variance), the result is a broadband elevation in power [40] (Fig 5A). Such broadband power elevations can be observed in the LFP [41] as well as intracellular membrane potentials of single neurons in awake macaque [42].

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Fig 5. The effect of varying simulated neural inputs on output spectra.

The effect of manipulating one of the three neural inputs used in the simulations produced different effects in the spectral power of the local field potential (LFP) of 200 neurons. (A) For C¹ (broadband), a high amplitude results in a broadband power elevation, with no narrow peaks in the spectrum. (B) For C² (gamma), a high correlation results in a narrowband gamma power elevation, with no broadband elevation or change in alpha power. (C) For C³ (alpha), a high amplitude input results in a narrowband power elevation in the alpha band, with no change in broadband power or narrowband gamma power. For each spectrum in each plot, 10 simulated trials were run. The plotted spectra are averaged across the 10 trials and are computed from I(t), the time series after leaky integration of the inputs. (Code to reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script05_Fig5.m).

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Gamma input (C²). Input C² consisted of band pass noise (40 Hz to 60 Hz), with fixed amplitude on all trials and with coherence across neurons that varied between trials. This input approximates the signals giving rise to narrowband gamma oscillations. Across different conditions, we varied the correlation between neurons of C² rather than the amplitude for individual neurons, which was fixed. This corresponds to a slice in the plots in Fig 2 such that the variance axis is fixed at a nonzero value. The motivation for this comes from empirical observations that large gamma oscillations in the LFP tend to reflect increased coherence between neurons [43, 44]. This is opposite to the broadband input (C¹), for which we varied the amplitude (variance) in individual neurons across trials, rather than the synchrony between neurons. Narrowband gamma oscillations with a peak between 30 Hz and 80 Hz can be observed in the LFP [45, 46] as well as in the membrane potential of individual pyramidal neurons [47]. When we increase the correlation of C² in our simulations, the result is an increase in the amplitude of the LFP in the gamma band (Fig 5B), much like narrowband gamma signals observed in microelectrode recordings [48] and human ECoG [38].

Alpha input (C³). The alpha input consisted of inhibitory oscillations at approximately 10 Hz, with fixed correlation between neurons and varying amplitude across conditions. This corresponds to a slice in the plots in Fig 2 in which the ρ-axis is fixed at a nonzero value. The oscillations were inhibitory, i.e., the mean was below 0 (compare C³ versus C¹ and C² in Fig 4). Because C³ was inhibitory, it resulted in less depolarization (or hyperpolarization in extreme cases), opposite the effect of C¹, which resulted in depolarization. This input approximates the signals giving rise to alpha oscillations (Fig 5C). Pyramidal neurons in visual cortex have been hypothesized to receive periodic inhibition, with pulses arriving at approximately 10 Hz [49, 50]. Individual neurons in visual cortex can indeed show subthreshold membrane oscillations at frequencies around 10 Hz [51].

2.3 Fitting the simulation parameters to ECoG responses and predicting BOLD data

The simulations were structured to approximate the experimental design and the results of our ECoG experiments. To match the design of our ECoG experiments, a simulated experiment consisted of 240 trials, each of which were 1 second long (30 repeats of 8 conditions). The LFP time series were transformed to power spectra, which were averaged across the 30 repeated trials of the same condition. The simulation parameters—i.e., the level of the three inputs, C¹, C², and C³—were fit to the measured ECoG summary metrics (broadband, gamma, and alpha) for each of the 8 conditions for a particular electrode (Fig 6). To verify the validity of this procedure, we asked whether the simulations using the fitted parameters produce simulated spectra, which, when analyzed like the ECoG spectra, reproduce the original values of broadband, gamma, and alpha. In other words, do we close the loop from measured spectral components (broadband, gamma, and alpha) to inferred input parameters (C¹, C², C³) to simulated population activity, to simulated spectral components (broadband, gamma and alpha)? The original values are not reproduced exactly because the simulations are stochastic, but overall, the original broadband, gamma, and alpha values are recovered with high accuracy (S9 Fig).

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Fig 6. Parameter fits for simulations.

(A) A function was fit between the C¹ input and the simulated broadband output (black line). Inverting this function allows us to take the measured electrocorticographic (ECoG) broadband values (green arrows to dots colored by stimulus condition) and convert these into estimates of the C¹ input levels (orange arrows). Because the simulations contain noise, the predicted broadband need not match the measured broadband exactly; however, the agreement is close, as shown in S9 Fig. The parameters for C² and C³ were fit similarly. (B) The C¹, C², and C³ parameters for the simulation of responses in a V2 electrode for 9 stimuli (8 contrast patterns plus a blank condition). These parameter fits were made using the full ECoG dataset (for this electrode), so there are no error bars on the input parameters. (Code to reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script06B_Fig6.m).

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As described above, the fitting of the parameters for C¹, C², and C³ was constrained by the assumptions that for C¹, the correlation between neurons was 0 (and the amplitude was varied for fitting); for C², the amplitude was fixed at a nonzero value (and the correlation was varied for fitting); and for C³, the correlation was fixed at a nonzero value (and the amplitude was varied for fitting). Results from alternative models with different constraints show poorer fits and are described briefly below and more extensively in S7 Fig.

Importantly, the parameter fits did not take into account the measured BOLD responses. Hence the simulations provided a test: if the input parameters were chosen to produce outputs that match the measured ECoG responses (training data), does the simulated BOLD signal accurately predict the measured BOLD signal (test data)? We measured BOLD responses in four healthy subjects to the same visual stimuli as used in ECoG (subjects are different from the ECoG subjects) and extracted the signal from regions of interest in visual cortex matched to the previously recorded ECoG electrode locations (S2 Fig and S3 Fig). For an example V1 site, the predicted BOLD signal accurately matched the measured BOLD signal, with 89% of the variance in the measured BOLD signal explained by the prediction, as quantified by R²—the coefficient of determination (Fig 7A). Across V1 sites, the predicted BOLD signal from the simulations accounted for a median of 80% of the variance in the measured data (Fig 7C). For an example V2 site, the predicted BOLD signal also matched the measured BOLD signal (R² = 0.74, Fig 7B). Across V2/V3 sites the simulations explained a median of 40% of the variance in the data. The explained variance in V2/V3 is substantial but lower than in V1. One likely reason for the higher variance explained in V1 is that for the particular stimuli used in these experiments (gratings and noise patterns), the BOLD response reliability was higher in V1. For example, the median R² computed by using half the BOLD data as a predictor for the other half (split half by subjects) was 86% for V1 and 63% for V2/V3. Similarly, the stimulus-evoked BOLD responses in V1 were larger than in V2 and V3, with more stimulus-related variance to explain: a mean of 1.8% signal change in V1 versus 1.2% in V2 and 0.8% in V3 (S4 Fig). It is possible that a stimulus set more tailored to extrastriate areas, such as textures or more naturalistic scenes, would have evoked more reliable responses in extrastriate cortex.

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Fig 7. Accuracy of predicted blood-oxygen-level dependent (BOLD) signals from simulated neuronal activity.

(A) Simulated BOLD (x-axis) versus measured BOLD (y-axis) for a V1 site. Each color corresponds to one stimulus condition (red dots, grating patterns; blue dots, noise patterns; black dot, uniform stimulus, or blank). Error bars indicate 68% confidence intervals, bootstrapped 100 times over 30 trials per stimulus for simulation and over repeated scans for BOLD data. (B) The same as A but for a V2 site. (C) The accuracy of BOLD predictions for all V1 and V2/V3 sites. Each site is indicated by a yellow dot. The orange lines show the medians and the red boxes the 0.25 and 0.75 quantiles. The thin, gray, solid lines show the BOLD data-to-data reliability, and the gray dashed lines show the accuracy when the BOLD data and trial conditions are shuffled in the training dataset. Accuracy is quantified as the coefficient of determination after subtracting the mean from the data and the predictions, and dividing each variable by its vector length. Because the simulations were fit to electrocorticographic (ECoG) data and tested on BOLD data, the predictions are cross-validated, and the coefficient of determination spans (−∞,1]. A value of −1 is expected when the data and predictions are unrelated and have equal variance, as in the case of the shuffled control analysis. (Code to reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script07.m).

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For each of the 22 simulations, the three input parameters C¹, C², and C³ defining each of the 8 stimulus conditions were fit to produce the LFP data from the corresponding ECoG electrode. By design, the C¹ (broadband) and C³ (alpha) inputs were fit to ECoG data by varying the level per neuron, whereas C² was fit to data by varying the correlation across neurons. In principle, for any of the three inputs, the ECoG data could have been fit by varying either the level per neuron or correlation across neurons. For completeness, we tested all 8 combinations of models (S7 Fig). The most accurate model, quantified as the R² between the measured BOLD and the simulated BOLD (median across the sites in V1 or in V2/V3), was the simulation type used in the main text, in which C¹ and C³ varied in the level per neuron and C² varied in the correlation across neurons. Models in which the broadband correlation rather than level was used to fit the ECoG broadband power were much less accurate. The models in which the gamma LFP power was fit by modulating the level rather than the correlation in the simulated population caused a small drop in R².

2.4 Relation between ECoG and BOLD in simulation and data

The previous analysis showed that when simulations were fit to ECoG data, the simulated BOLD response predicted the measured BOLD response. Here, we used regression analyses to assess how the simulated LFP predicted the simulated BOLD response and how the measured LFP predicted the measured BOLD response.

2.4.1 Examples of the relation between the BOLD amplitude and LFP features.

We first consider an example V1 site (Fig 8A: same site plotted in Fig 7A). The BOLD amplitude for the different stimuli was accurately predicted by the broadband response (R² = 0.85) but not by the narrowband gamma or alpha power (R² = −0.06, R² = −0.07, respectively). Hence, in V1, only broadband power was a good predictor of BOLD amplitude (indicated by the black outlines in Fig 8). Because there were only 8 data points to fit in each of the 3 correlations, we used cross-validation to avoid overfitting: linear regression was used to fit the BOLD signal to the ECoG measure in separate halves of the data, and the R² was computed from the left out half of each dataset.

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Fig 8. The accuracy of predicted blood-oxygen-level dependent (BOLD) signals from electrocorticographic (ECoG) components.

The correlation between ECoG and BOLD was calculated for a V1 site and a V2 site. The locations of one sample electrode in V1 and one in V2 are indicated by the enlarged white discs on the cortical surface for subject 1. (A) In a foveal V1 site, the broadband ECoG amplitude accurately predicted the BOLD signal (left). Error bars show 68% confidence intervals across bootstraps. Narrowband gamma power (center) and alpha power (right) were uncorrelated with BOLD. (B) In a V2 site, the broadband ECoG was weakly correlated with BOLD (left). Narrowband gamma did not predict BOLD (middle). Alpha was negatively correlated with BOLD (right). Scatter plots for all other V1 and V2/V3 sites are shown in S5 Fig. (C-D) The same as A and B but for simulated neuronal population data fit to the V1 and V2 ECoG data. For all panels: data points are the bootstrapped median across 30 trials per stimulus (ECoG) and across scans (BOLD). The trend lines are least square fits to the 8 data points plotted. The R² values are the coefficient of determination computed by cross-validation, with a regression fit to half the data and evaluated on the other half. The black outlines indicate the regressions that show reliable predictors of the BOLD signal—broadband in V1, broadband and alpha in V2/V3. (Code to download data and reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script09A_Fig8AB_Fig9AB.m and function ns_script07B_Fig7AB_Fig8CD.m).

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A different pattern was found in the V2/V3 data. For an example V2 site (Fig 8B: same site plotted in Fig 7B), the broadband power did not predict the BOLD signal as accurately as it did for the V1 site (R² = 0.31), whereas the alpha power predicted the BOLD signal more accurately than it did for the V1 site (R² = 0.59). Gamma power did not predict the BOLD signal, similar to the V1 data.

For the simulation fit to the ECoG data from the example V1 electrode, the relation between the BOLD signal and the LFP (Fig 8C) was similar to the measured V1 data: the LFP broadband response predicted the BOLD signal (R² = 0.89), whereas the power of narrowband gamma oscillations and alpha oscillations did not (R² = 0.01). In this simulation, similar to the data, the LFP broadband response was the only good predictor of the BOLD signal. Hence, the data and the simulation match in that they both show that some features of the LFP are good predictors of BOLD (broadband), and some are poor predictors (alpha and gamma).

For the simulation fit to the ECoG data from the example V2 electrode, the relation between the BOLD signal and the LFP (Fig 8D) was similar to the measured V2 data. First, the broadband LFP was again a good predictor of the BOLD signal, and gamma power was again a poor predictor. Second, the power of alpha oscillations had a strong negative correlation with BOLD (R² = 0.73).

In the simulations, the correlation between broadband and BOLD is higher for V1 than for V2, and the correlation between alpha and BOLD is higher for V2 than for V1. These differences were not due to a difference in the types of inputs (C¹, C², C³), nor to the way BOLD or LFP was derived from the population activity—the simulation algorithm and the analysis code were identical for all simulations. The difference arises only from the different parameters—that is, there were different mixtures of C¹, C², and C³ for the V1 and the V2 simulations. This highlights the fact that the identical mechanism converting neural activity to BOLD (“neurovascular coupling”), modeled here as power in the time series summed across neurons (Eq 3), can produce very different correlations between BOLD and features of the LFP, depending on the neural activity.

2.4.2 The relation between the BOLD amplitude and LFP features across sites.

In the example V1 site and the corresponding simulation, the BOLD signal was well predicted by broadband increases (Fig 8B and 8D). In the example V2 site and the corresponding simulation, the BOLD signal was predicted by both broadband increases and alpha decreases (Fig 8C and 8E). By explicitly modeling both the population response and the population-to-instrument transformations, we see that a difference in the relation between instrument measures (BOLD and ECoG) can arise from a difference in the population response, without a difference in neurovascular coupling. We now ask (1) whether these effects are consistent across the measured V1 and V2/V3 sites and (2) how a multiple regression model using broadband, gamma, and alpha as predictors fits the BOLD response for both data and simulation.

As we argued in the Introduction, we believe there is no single, general transfer function that can predict the BOLD signal from the LFP. Yet a regression model linking the two measures can be a useful way to summarize the results of a particular experiment/simulation and to compare results between different experiments/simulations. Here, we fit several regression models to each simulation and to the data (Fig 9). The regression models predicted the simulated or measured BOLD response from either a single LFP component (broadband power, gamma power, or alpha power) or from combinations of LFP components (each of the pairwise combinations, and the three components together). These regression models were fit separately for each of the 9 sites in V1 and each of the 13 sites in V2/V3 and for the 22 corresponding simulations. The accuracy of each model was assessed by the variance explained in the cross-validated data (R², the coefficient of determination). With a cross-validation procedure, there is no advantage in accuracy for models with more free parameters, and accuracy is reduced rather than increased by overfitting. The cross-validated R² was compared to the R² in a null distribution, derived from shuffling the assignment between data and stimulus conditions.

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Fig 9. Explained variance in the blood-oxygen-level dependent (BOLD) functional magnetic resonance imaging (fMRI) signal in the simulations and in data.

(A) Variance in the measured BOLD signal explained by broadband, gamma, and alpha changes in the electrocorticographic (ECoG) data. The colored box plots show the variance explained by each of the 7 model types: black bar = median, upper and lower boxes = quartiles, and error bars = data range excluding outliers (outliers plotted as red plusses). The R² was cross-validated (split between subjects for BOLD and stimulus repetitions for ECoG) to ensure that the R² can be compared between models with different numbers of explanatory variables. Gray dashed lines indicate the noise floor (R² when shuffling the BOLD labels in the training data), and the solid gray lines indicate the data-to-data reliability for the BOLD signal. Bottom: the regression coefficients show whether the broadband, gamma, and alpha signals were positive or negative predictors of the BOLD signal. A red * in the lower plot indicates whether regression coefficients differed significantly from 0 by a bootstrap test (p < 0.05). (B) The same as A but for the 13 V2/V3 electrodes. (C) The same as A but for the 9 simulations fitted to V1 data. The R² was cross-validated (split between even and odd stimulus repetitions). D) Same as C, except for the 13 simulations fitted to the V2/V3 data. (Code to download data and reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology).

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Data. The single-parameter regression models across V1 electrodes (Fig 9A) show the same pattern as the single electrode examples in Fig 8: broadband alone was a good predictor of BOLD (median cross-validated R² = 0.70 across 9 sites), while gamma and alpha alone were not (gamma R² = 0.04, alpha R² = 0.04). Across all regression models, the observed BOLD signal was best predicted by a combination of broadband and alpha changes with an R² = 0.78, close to the data-to-data reliability (R² = 0.82). Adding gamma as a predictor to either the broadband-only model (model 2 versus model 1) or to the combined broadband and alpha model (model 7 versus model 5) did not increase model accuracy, confirming that the gamma amplitude was not predictive of the BOLD signal.

The single-parameter regression models across V2/V3 sites (Fig 9B) show that alpha power predicted the BOLD signal (R² = 0.32, with a negative Beta value), whereas broadband alone was only slightly more accurate than a control, shuffled model, and gamma alone had no predictive power. As in V1, the BOLD response was best explained by a regression model combining broadband and alpha (R² = 0.39; see also S6 Fig) or a model using all three predictors (R² = 0.42). Overall, compared to V1, the BOLD signal in V2/V3 was less accurately predicted by the regression models based on the electrophysiological measurements. As with the case of predicting BOLD from simulated neuronal activity, predicting BOLD from ECoG measures is limited by the reliability of the stimulus-evoked BOLD signal, which was lower for V2/V3 than for V1 (0.59 versus 0.82 R²).

Simulations. For the simulations, we expect broadband power to positively predict the BOLD signal and alpha power to negatively predict the BOLD signal because of the construction of the simulations: broadband and alpha power elevations were achieved by increasing the variance per neuron, rather than correlations between neurons; the converse was true for gamma. Nonetheless, solving the regression models can be informative because, as seen in Fig 8, simulations with the identical input types and the identical analysis can lead to different patterns, depending on the parameters (weights) in the simulations. Moreover, the regression analyses of the simulated data can be compared against the regression analyses of the measured data.

The results from the regression model on simulated V1 LFP and BOLD (Fig 9C) were qualitatively similar to the V1 data: broadband alone was a good predictor of BOLD (median cross-validated R² = 0.70 across 9 simulations), while gamma and alpha alone were not (median R² = 0.04 for both predictors). For simulations fit to V2/V3 ECoG data (Fig 9D), alpha alone predicted the BOLD signal with moderate accuracy (median R² = 0.32).

Similar to the data, the BOLD response in the simulations fit to V1 and V2/V3 was well explained by a regression model combining broadband and alpha (R² = 0.78, R² = 0.39, Fig 9C and 9D). The regression coefficients for these models were positive for broadband and negative for alpha. A model that incorporated all three LFP measures—broadband, alpha, and gamma—explained little to no additional variance for either simulation, confirming our earlier observation that narrowband gamma power was not correlated with BOLD amplitude in simulated data. The generally higher variance explained in V1 than in V2/V3 again matches the higher BOLD reliability in V1 for these experiments.

Across simulations and datasets, a general pattern emerges. The broadband signal was positively predictive of BOLD, and alpha power was negatively predictive of BOLD. Narrowband gamma had no consistent relation with BOLD. While the relationships between broadband and BOLD and between alpha power and BOLD were consistent in terms of sign (the former positive, the latter negative), the level was not always the same. As we noted in the example sites shown in Fig 8, and the summary across sites in Fig 9, the broadband power was more strongly predictive of BOLD in V1, and alpha power was more strongly predictive in V2/V3. An examination of responses to the different stimulus types clarifies the difference between V1 and V2/V3 in these data. Specifically, the BOLD response in V2/V3 to noise patterns was underpredicted by the broadband response alone (S5 Fig). In V2/V3, alpha decreased more for the noise patterns, and this alpha decrease accounted for the BOLD change unexplained by broadband (S1 Fig). This helps to explain why a model that includes broadband and alpha is much more accurate for V2/V3 than a model that includes only broadband. In contrast, for V1, the BOLD response was well predicted by broadband power in most sites, with little systematic prediction failures and little room for increased model accuracy when adding predictors such as alpha power.

2.4.3 Correlation between BOLD and LFP across all frequencies.

In the previous section, we modeled the BOLD responses as a linear function of three components derived from the LFP. These features—broadband power, narrowband gamma power, and narrowband alpha power—are summary metrics of the power spectrum. We also tested how the power at each frequency in the ECoG data and in the simulated LFP correlated with the BOLD response (measured and simulated, Fig 10).

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Fig 10. The correlation between blood-oxygen-level dependent (BOLD) and local field potential (LFP) as a function of frequency.

(A) The correlation between electrocorticographic (ECoG) and BOLD for the V1 data shows a positive correlation between ECoG and BOLD for a broad range of frequencies, except those including the alpha changes. Gray lines represent the 9 individual V1 electrodes, the black line is the average, and the red line corresponds to the example sites shown also in Fig 7. (B) In the V2/V3 data, there was a strong negative correlation between ECoG and BOLD in the alpha range around 10 Hz and a positive correlation between ECoG and BOLD for a broad range of frequencies. Gray lines represent the 13 individual V2/V3 electrodes, the black line is the average, and the red line corresponds to the example electrode shown also in Fig 7. Note that neither the V1 electrodes nor the V2/V3 electrodes show a peak at the gamma frequency. (C) The correlation between LFP and BOLD for simulations fit to V1 shows that there is a positive correlation across most frequencies, except those including the alpha and gamma changes. (D) The correlation between LFP and BOLD for the simulations fit to V2/V3 shows that there is a strong negative correlation around 10 Hz and a positive correlation across a broad range of frequencies. (Code to download data and reproduce this figure can be found on https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology function ns_script10_Fig10.m).

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Data. We calculated ECoG power for each frequency from 1 Hz to 200 Hz and correlated the power changes from baseline with BOLD changes from baseline. In V1, ECoG responses across all frequencies except the alpha band were positively correlated with the BOLD response (Fig 10), which is consistent with the regression analyses of the summary metrics, showing that broadband ECoG power was the best predictor of the BOLD signal.

The pattern of correlation between ECoG power and BOLD in V2/V3 was similar to that found in V1, although the overall level of correlation was lower (Fig 10). There were positive correlations between ECoG and BOLD extending across most frequencies that were weaker than in V1, and there was a negative correlation for most sites in the alpha band.

Simulation. The correlation between simulated BOLD and ECoG power in V1 was qualitatively similar to that found in the data. In simulations fit to V1 ECoG data, the LFP correlated well with the BOLD signal across all frequencies except those in the alpha band (8 Hz to 15 Hz) and below, and those in the gamma band (40 Hz to 60Hz).

In simulations fit to V2/V3 ECoG data, the pattern was similar, except that the correlation was negative in the alpha band rather than 0, and weaker but still positive in the rest of the spectrum. These patterns match the summary metrics of alpha, gamma, and broadband shown in Fig 9.

Site to site differences. There were some differences between V1 sites. For example, in two sites, the correlation across frequencies dipped in the gamma band (30–80Hz), similar to simulated data. These are also the two sites that showed the largest amplitude gamma responses (sites 8 and 9 in S5 Fig). In other words, when cortical sites showed large gamma signals, these signals were uncorrelated with BOLD. The fact that in 7 V1 sites there was a positive correlation between BOLD and LFP power spanning 30 Hz to 80 Hz might seem inconsistent with our earlier observation that narrowband gamma power was not predictive of the BOLD signal in V1 sites (Fig 8 and Fig 9). However, in this analysis, the narrowband and broadband power are not modeled separately, and the positive correlation between power at 30 Hz to 80 Hz in Fig 10 thus likely suggests that broadband power extends into this band, since broadband changes can extend across all frequencies [11, 40]. Therefore, if there is little to no narrowband response, we would expect a positive correlation between BOLD and ECoG throughout all frequencies.

There were some site-to-site differences in the correlation between alpha and BOLD. For example, some sites showed a positive correlation with BOLD in the alpha range, and others showed a negative correlation (Fig 10A and 10B). These site-to-site differences depend on the range of responses evoked by stimuli. For example, for electrodes in which stimuli evoked a large range of power changes in the alpha band, alpha was more strongly correlated with BOLD. Similarly, for electrodes in which stimuli evoked a large range of broadband responses, broadband was more highly correlated with BOLD (S8 Fig). This pattern did not hold for narrowband gamma power changes.

3.1 Changes in broadband power predict BOLD

Many studies have reported correlations between BOLD and power in the gamma band LFP (30–130 Hz) (review for human studies: [52]). Yet, changes in gamma band power do not reflect a single biological mechanism. For example, several recent studies have emphasized that LFP power changes in the gamma band reflect multiple distinct neural sources, including narrowband oscillations and broadband power shifts, with very different stimulus selectivity and biological origins [38, 53, 54].

Broadband changes have been proposed to reflect, approximately, the total level of Poisson-distributed spiking (or spike arrivals) in a local patch of cortex [40]. In contrast, the narrowband gamma response is caused by neural activity with a high level of cell-to-cell synchrony [55] and likely depends on specialized circuitry [56]. While the two responses are sometimes distinguished as “high gamma” (referring to broadband signals) and “low gamma” (referring to oscillatory signals), this distinction is not general. Broadband signals can extend into low frequencies [11, 57] so that the two signals can overlap in frequency bands. Hence, separating gamma band field potentials into an oscillatory component and a broadband (nonoscillatory) component is not reliably accomplished by binning the signals into two temporal frequency bands, one low and one high, but rather requires a model-based analysis, such as fitting the spectrum as the sum of a baseline power law (to capture the broadband component) and a log-Gaussian bump (to capture the oscillatory component) [38].

There is strong experimental support for the idea that increases in broadband LFP power primarily reflect increases in asynchronous neural activity rather than increases in coherence. First, experiments have shown that broadband power is correlated with multiunit spiking activity [54, 58]. Second, unlike the case of narrowband gamma LFP, changes in broadband LFP are not accompanied by changes in broadband spike-field coupling (Figure 1A-B in [43]). The possibility that neuronal synchrony sometimes affects broadband signals cannot be ruled out, for example, as shown in cases of pharmacological manipulations in nonhuman primates [59]. In such cases, there would not be a simple relationship between broadband power and BOLD.

The prior literature has not shown definitively whether broadband LFP, narrowband gamma, or both predict the BOLD signal. The first study that directly compared simultaneously recorded BOLD and electrophysiology showed that both LFP power in the gamma frequency range (40–130 Hz) and multiunit spiking activity (MUA) predicted the BOLD signal [16] and further, that when the LFP power diverged from MUA, the gamma band LFP predicted the BOLD signal more accurately than did spiking. This study however did not separately test whether a narrowband (oscillatory) or a broadband (nonoscillatory) component of the LFP better predicted the BOLD response.

Other studies, too, have shown a variety of patterns when correlating LFP power changes in the gamma band with BOLD. Some reported that BOLD amplitude correlates with narrowband gamma activity [13], while others showed that BOLD correlates with broadband changes [11], and many did not distinguish narrowband from broadband power in the gamma band [60]. Simultaneous recordings of hemodynamic and neuronal activity in macaque V1 showed that BOLD signals from intrinsic optical images can occur in the absence of gamma band LFP changes [61] and that, in some circumstances, multiunit activity predicts the BOLD signal more accurately than gamma band LFP [24, 62].

Here, we separately quantified the broadband power (spanning at least 50–150 Hz) and narrowband gamma power. We found that the amplitude of broadband changes accurately predicted the BOLD signal in V1. The empirical results and the models help resolve the question of why “high gamma” has been shown to correlate with BOLD, and “low gamma” sometimes does not [24]. The likely reason is unrelated to the difference in frequency range, nor to the size of the spectral perturbation in the LFP. In fact, the elevation in broadband power is relatively small (2- or 3-fold) compared to the elevation in power often observed in narrowband gamma oscillations (10x or more)[38]. Instead, “high gamma” is predictive of the BOLD signal in many cases not because of the specific frequency range, but because this signal captures the level of asynchronous neuronal response; this signal happens to be most clearly visible in the high-frequency range (>100 Hz) in which it is not masked by rhythmic lower-frequency responses. Hence, the distinction in predicting the BOLD response is not about “high” versus “low” gamma but rather synchronous versus asynchronous responses, and the broadband signal, sometimes labeled high gamma, maps onto the first term on the right-hand side of Eq 4, the portion of the field potential measurement that sums the energy demand of each neuron.

Our model fits and data support this view. When we captured the stimulus-related broadband response by simulating a change in broadband coherence across neurons rather than a change in the level of response in each neuron, our predicted BOLD response was highly inaccurate (S7 Fig).

3.2 Changes in narrowband gamma power do not predict BOLD

In contrast, we propose that “low gamma” often does not predict the BOLD response because “low gamma” reflects narrowband oscillatory processes, which largely arise from a change in neuronal synchrony across the population rather than a change in the response level per neuron. This corresponds to the second term in the right-hand side of Eq 4, the portion of the field potential measurement that reflects the cross-power arising from currents in different neurons, which in our model, is independent of the signals giving rise to the BOLD signal.

Our results and model do not argue that narrowband gamma oscillations will never be predictive of the BOLD signal. If, in a particular experiment, narrowband gamma oscillations were to covary with broadband increases, we would expect both signals to correlate with BOLD. This might occur in an experiment with gratings of different contrast; with increasing contrast, narrowband gamma responses, broadband responses, and BOLD responses all increase [21, 63], and all three measures would correlate across stimuli. In such an experiment, if narrowband gamma oscillations had a higher signal-to-noise ratio than the broadband response, then the oscillatory signal would likely show a higher correlation with BOLD. In contrast, when the choice of stimulus or task can independently modulate broadband power and gamma oscillations so that the two LFP measures are not correlated, as in the experiments presented here and previously [38], then gamma oscillations will not strongly correlate with BOLD.

Our simulation and empirical results are consistent with studies that varied chromatic contrast and spatial frequency while measuring MEG and BOLD. These studies found that BOLD and narrowband gamma activity did not match in stimulus specificity [18, 19]. It is likely that these stimulus manipulations, like ours, independently modulated narrowband gamma power and broadband power, although the studies did not quantify broadband fields, which are more challenging to measure with MEG than with ECoG [64]. We speculate that broadband fields spanning the gamma range would have shown a higher correlation with BOLD.

3.3 Neuronal synchrony and the BOLD signal

In our model, the LFP measures are highly sensitive to neuronal synchrony, whereas BOLD is not. In our simulations, increases in neuronal synchrony drove narrowband gamma oscillations in the field potential. There are other cases of population activity with a high degree of neuronal synchrony. One example is the steady-state evoked potential associated with a periodic stimulus [65, 66]. Previous studies have described discrepancies between evoked potentials and the BOLD signal, such as in the case of spatial summation [11], directional motion selectivity [7, 8] and spatial attention [9, 10]. Our modeling framework suggests that the neural sources generating the steady-state potential (synchronous neural activity) are likely to be only weakly related to the BOLD signal (depending largely on asynchronous signals), as these sources will primarily affect the second term on the right-hand side of Eq 4 (cross-power). This does not imply that the two measures are always or even usually discrepant; the BOLD signal and steady-state potentials are likely to correlate any time that the steady state signals correlate with other measures of neural activity. When measures do dissociate, we do not conclude that one measure is more accurate; instead, the measures offer complementary views of the population activity, emphasizing the degree of synchrony or the average level of the response. An intriguing question is how each of the two signals contributes to perception and behavior [67].

Neural synchrony can also emerge without being time-locked to the stimulus, often called “induced synchrony” or “induced oscillations” [68]. In our simulation, we assumed that narrowband gamma LFP changes were induced by increases in synchrony between neurons and not by changes in the level of gamma power within the individual neurons. In contrast, we assumed that broadband LFP increases were induced by increased broadband activity in individual neurons and not by increased broadband coherence between neurons. (In Eq 4, a change in the left-hand side, LFP power in the gamma band, can be produced by a change in either the first or second term on the right). This explains why, in our simulation, the broadband power was correlated with BOLD, whereas the LFP gamma power was not, findings that were also confirmed by the data. Were our assumptions justified?

In principle, an increase in narrowband gamma power in the LFP could arise because the neurons synchronize in the gamma band or because ongoing gamma oscillations within each neuron increase in amplitude, independent of coordination between neurons. There is strong experimental support for the former. Experiments that measure both intracellular membrane potential from single neurons and the extracellular LFP show that when there is an increase in narrowband LFP gamma power, the gamma power from individual neurons becomes more coherent with the LFP [47]. Moreover, the coherence between local spiking and the LFP also increases in the gamma band when LFP gamma power increases [43]. These results are consistent with our assumption that a significant part of the increase in gamma LFP power arises from a change in population coherence. To our knowledge, it is not certain whether there is also some increase in the level of gamma signals within individual neurons when the narrowband gamma band LFP power changes. However, since we can attribute a large part of the change in gamma LFP to a change in coherence, we infer that we can only attribute, at most, a small part of the change in gamma LFP to the level of gamma power within neurons.

In our simulation, we made two simple but extreme assumptions. First, we assumed that gamma oscillations occur with no change in the total level of neural activity, and hence no change in metabolic demand or BOLD. Second, we assumed that broadband responses occur with no change in neural synchrony. While these assumptions are likely incorrect at the limit, the simulations nonetheless captured the pattern of ECoG and fMRI results obtained in our datasets. Alternative models in which the broadband response was caused by a change in synchrony were much less accurate (S7 Fig). Models in which gamma responses were caused by a change in level were only slightly less accurate and cannot be ruled out entirely (S7 Fig). However, the regression models fit to our data (Fig 9) show that the power of narrowband gamma oscillations does not predict the BOLD response. Hence the most parsimonious explanation is that these responses in the LFP are caused in large part by changes in synchrony.

3.4 Mean offsets and the BOLD signal

Both our measurements and our simulations showed that broadband electrophysiological responses were related to, but did not fully account for, the BOLD signal. This was especially evident in Simulation 2 and extrastriate data (V2/V3). In these cases, the amplitude of low-frequency oscillations (8–15 Hz) was negatively correlated with the BOLD signal, independent of broadband signals. Numerous previous studies have reported that low-frequency oscillations are anticorrelated with BOLD, including measurements in motor, visual, and language areas [20–22, 69–71]. This result may appear to conflict with the prior discussion, since we argued that oscillations (to the degree that they reflect neuronal synchrony) should have little to no effect on metabolic demand or the BOLD signal. It is therefore important to ask why low-frequency oscillations sometimes correlate with the BOLD signal, both in data and in simulation.

One explanation is that alpha oscillations, or a mechanism that generates the oscillations, affect the BOLD signal indirectly by inhibiting cortical activity. According to this explanation, an increase in alpha power results in a decrease in local spiking activity, in turn reducing metabolic demand and the BOLD signal [72]. Alpha oscillations may indeed co-occur with reduced cortical excitation [73]. However, if this coupling between alpha power and spiking was the only explanation for the relationship between alpha power and BOLD, then a more direct measure of neuronal excitation, such as broadband or multiunit activity, would adequately predict the BOLD signal; alpha power would negatively correlate with the BOLD signal but would provide no additional predictive power. Our data and model do not support this explanation, as we find that for most cortical sites, the most accurate predictor of the BOLD signal is a combined model including both the amplitude of alpha oscillations and broadband power.

We therefore propose that in addition to the indirect effect of modulating cortical excitability, alpha oscillations are also accompanied by a mean shift in membrane potential, making it less depolarized (i.e., closer to the equilibrium potential), and this shift reduces metabolic demand. Indirect evidence for a mean shift comes from MEG and ECoG studies [49, 50, 74], which refer to alpha oscillations as being asymmetrical (i.e., they are not centered at 0—there is a mean shift). This can be explained by a simple process: if alpha oscillations reflect periodic inhibitory pulses, then on average, they will cause a hyperpolarization (or less depolarization). If the neuron was slightly depolarized before the inhibitory alpha pulses, then the pulses would push the neuron toward equilibrium and hence a lower-energy state. In this view, large alpha oscillations reflect larger inhibitory pulses, reducing depolarization. We suggest that this reduced depolarization affects metabolic demand in two ways: by reducing spiking (as discussed above) and by maintaining a less-depolarized state, reducing metabolic demand. In our model, the contribution to the BOLD signal from each neuron is the power in the time series (Eq 3), and the mean contributes to power. The idea that a mean shift in the membrane potential affects metabolic demand (in addition to altering excitability) is consistent with the observation that slowly changing currents (<0.5 Hz) correlate with BOLD fluctuations [12, 75]. Moreover, if alpha oscillations are associated with a mean shift in membrane potential, this would explain why cortical excitability depends on the phase of the alpha cycle: at one phase, the membrane potential is more depolarized, and hence cortex is more excitable, and in the opposite phase, cortex is more hyperpolarized and hence less excitable. This is consistent with the observations that the threshold for eliciting a phosphene with TMS changes with alpha phase [76, 77] and that the alpha phase at the time of stimulus presentation influences the size of the BOLD response in visual cortex [78].

Inhibition takes two neurons—one to inhibit and one to be inhibited. In our simulations, the alpha oscillations (C³) were associated with inhibitory fluctuations in the membrane potential (mean below 0), which in turn was associated with decreases in BOLD. It is important to note that these fluctuations are meant to capture the effect of local inhibition on the postsynaptic neurons (the neurons being inhibited). The inhibitory neurons themselves are presynaptic, and the action of inhibiting other neurons is presumably an active process that consumes energy. Therefore, inhibition is expected to increase energy demand in some neurons (the presynaptic neurons) and decrease energy demand in other neurons (postsynaptic neurons). We did not model the inhibitory neurons explicitly; however, the neural activity associated with active inhibition would be expected to contribute to the measured broadband signal in the ECoG data and is implicitly included in the broadband inputs in our simulations (C¹). More complex models (see paragraph 3.6) in which the circuitry of excitatory and inhibitory neurons is explicitly represented (such as [63, 79, 80]) may provide insight into how the balance between excitation and inhibition influences the field potential and the BOLD signal.

3.5 A single modeling framework accounts for patterns of LFP/BOLD correlations across sites

We found that the relationship between the BOLD signal and features of the ECoG data differed across cortical areas. For example, broadband changes in ECoG responses explained more variance in the BOLD data in V1 than in V2/V3. Conversely, low-frequency power decreases (alpha, 8–13 Hz) explained more variance in the BOLD signal in V2/V3 than in V1. In the absence of a model, we might have interpreted the results as evidence that neurovascular coupling differs across sites. Many previous studies have reported differences in the relation between LFP and BOLD as a function of site or condition, for example, between cortical and subcortical locations [81], across cortical regions [82, 83], between cortical layers [84], and as a function of medication [85]. Here, we showed that a difference in the relationship between LFP and BOLD need not arise because of a difference in neurovascular coupling. In our results, Simulations 1 and 2, like V1 compared to extrastriate areas, showed differences in the relationship between LFP and BOLD, yet we used the identical model of neurovascular coupling in all simulations. The systematic differences in the two simulations arose because of differences in the neuronal population activity, not because of differences in neurovascular coupling. While our results do not exclude the possibility of differences in neurovascular coupling across locations or states, they do caution against interpreting differences in the relationship between field potentials and BOLD as evidence for a difference in neurovascular coupling, since they show that a single model can account for a variety of patterns. More generally, the V1 versus V2/V3 discrepancies bolster the argument that one cannot predict the exact relationship between BOLD and field potentials without also specifying the neuronal population activity.

3.6 The role of a simple model in understanding the relation between BOLD and LFP

A complete description of the biophysical processes giving rise to the BOLD signal and the field potential is far beyond the scope of this paper and is likely premature given the enormous complexity in the nervous system, the vascular system, and the coupling mechanisms between them. Instead, the purpose of our modeling framework was to first begin with a general principle, namely that BOLD and field potentials sum neural activity according to a different sequence of operations; second, to instantiate this principle in simple mathematical rules; third, to combine these rules with a minimal model of neural population activity; and finally, to ask to what extent such a model can account for the patterns in our data.

Our model omits many biophysical components, such as compartmentalized neurons, multiple cell types and vessel types, neurotransmitters, the dynamics of blood flow, and so on; hence, it is not a detailed simulation of the nervous system or vascular system. We modeled the BOLD signal as a function of dendritic currents summed across neurons within an imaging region. The logic motivating this is straightforward. Neuronal activity consumes a large amount of energy, and this energy demand is dominated by the cost of restoring the membrane potential following ionic flows from synaptic potentials and action potentials [86, 87]. As a result, the increased energy demand from neuronal responses is related to the dendritic currents. Neurovascular coupling is the process of increasing blood flow to meet this energetic demand; a failure of the hemodynamic response such as stroke can cause neuronal and glial cell death, highlighting the importance of the relationship between blood supply regulation and neuronal activity [33].

We modeled the hemodynamic response as being proportional to the energy demand from dendritic currents. This model was proposed as a computational summary of the approximate relationship between the BOLD signal and neuronal activity, not as a hypothesis about a causal mechanism. Recent work suggests that energy consumption per se (e.g., the change in the cerebral metabolic rate of oxygen consumption) is not the triggering mechanism for the increased blood flow, rather neurotransmitters and other molecules associated with synaptic events are part of a complex cascade that causes vasodilation and changes in blood flow [33, 88, 89]. The exact biological mechanism responsible for neurovascular coupling is an area of highly active, ongoing research [35]. The key assumptions in the model—that the BOLD signal is correlated with changes in membrane potential and that the order of operations differs for the BOLD signal and the LFP—makes accurate predictions for our dataset. A separate and important research question is how closely the biophysical mechanisms match this computational-level description, and what these mechanisms are.

The simplicity of the model has benefits. It facilitates an understanding derived from basic principles, similar to the advantages in building computational, rather than biophysical, models of neural responses [90–93]. Both types of models and empirical studies are valuable. Here, we emphasize that even with a highly simplified model of the BOLD signal, the field potential, and neuronal population activity, we are able to reconcile a wide range of findings in a complicated and technical literature. The model accounts for differences in how broadband field potentials and gamma oscillations relate to the BOLD signal. It can explain differences between cortical areas in the relationship between field potentials and BOLD. The model also provides an explanation for why the amplitude of alpha rhythms is negatively correlated with BOLD, even after accounting for the relationship between broadband signals and BOLD.

We note that drastic simplifications are the norm in many fields of neuroscience, such as receptive field modeling of visual neurons; most such models omit fixational eye movements, optical properties of the eye, retinal and cortical circuitry, etc., instead modeling responses as a few simple mathematical computations of the stimulus (filtering, thresholding, and normalization) [94]. These highly simplified models will certainly fail under some conditions [95], yet they have proven to be of immense value to the field [93], in part due to their simplicity and in part because the alternative in which the responses of visual neurons are computed from a complete, neurobiologically realistic model of the nervous system simply does not exist.

3.7 Reproducible computations

To test competing computational theories about the relation between the visual input, the LFP, and the BOLD response, it is essential to make sample data and code available for others [38, 53]. Following standard practices of reproducible research [96–98], the Matlab code of the simulation and sample data and code to reproduce the Figs in this manuscript can be downloaded at https://github.com/dorahermes/Paper_Hermes_2017_PLOSBiology.

3.8 Conclusions

To understand how the electrophysiology and BOLD responses are related, it is necessary to specify both the manner in which population activity transfers to the two signals and the neuronal population activity itself. The former shows that the covariance between neuronal time series has a large influence on the field potential and not the BOLD signal. Based on our simulations and empirical results, we made several inferences about the neuronal population responses mediating the BOLD signal and the LFP: that narrowband gamma oscillations in visual cortex likely arise more from synchronization of neural responses than a change in level of the neural response and hence have a large influence on the field potential and little influence on the BOLD signal, that responses that are asynchronous across neurons manifest in broadband field potentials and an elevated BOLD signal, and that low-frequency oscillations observed in field potentials are likely accompanied by a widespread hyperpolarization, which in turn reduces metabolic demand and the BOLD signal. Our model-based approach brings us a step closer to a general solution to the question of how neural activity relates to the BOLD signal.

4.1 Ethics statement

Informed, written consent was obtained from all subjects. The fMRI protocols were approved by the New York University IRB and ECoG protocols were approved by the Stanford University IRB, according to the principles expressed in the Declaration of Helsinki.

4.2 Simulated neuronal time series

Simulations were computed for a population of 200 neurons. Each simulation trial was 1 second long with millisecond sampling. The time series for each neuron was derived by summing three inputs, each 1 second long, followed by leaky integration with a time scale of 10 milliseconds to simulate temporal integration in the dendrite (Fig 4). Each simulation was fit to ECoG data from one electrode and consisted of 240 trials, 8 repeats of 30 stimulus conditions. A condition in the simulation was defined by the parameter settings for the three inputs (Fig 4): C¹ (broadband), C² (gamma), and C³ (alpha). Variations in these three inputs resulted in power changes in the broadband, gamma, and alpha LFP. The inputs were fit to data such that the simulated LFP power changes matched the ECoG data power changes for a particular electrode and stimulus.

4.3 Stimuli and task

Stimuli for ECoG experiments were reported previously [38]. In brief, for one subject, the stimuli came from 8 classes of patterns (30 exemplars per class, 20x20°), including high contrast vertical gratings (0.16, 0.33, 0.65, or 1.3 cycles per degree square wave) noise patterns (spectral power distributions of k/f⁴, k/f², and k/f⁰), and a blank screen at mean luminance (S2 Fig). For the second ECoG subject, there were the same 8 classes as well as two other stimulus classes–a high contrast white noise pattern and a plaid at 0.65 cpd. The fMRI subjects had the same 10 stimulus classes as the second ECoG subject.

4.4 ECoG procedure

ECoG data were measured from two subjects who were implanted with subdural electrodes (2.3 mm diameter, AdTech Medical Instrument Corp) for clinical purposes at Stanford Hospital. Informed, written consent was obtained from all subjects. ECoG protocols were approved by the Stanford University IRB. In 22 electrodes in V1 V2 and V3, broadband and narrowband gamma responses were quantified as before [38], and alpha power changes were calculated.

4.5 fMRI procedure

fMRI data was measured from four subjects (three female, ages 22–42) with normal or corrected-to-normal vision at the Center for Brain Imaging at NYU. Informed, written consent was obtained from all subjects. The fMRI protocols were approved by the New York University IRB. fMRI data were preprocessed and analyzed using custom software (http://vistalab.stanford.edu/software). Disc regions of interest (ROIs) (radius = 2 mm) were defined in fMRI subjects to match the position of the electrodes in ECoG subjects using a combination of anatomy, pRF centers, and visual field maps. The similarity between the ROI position and electrode position was compared via visual inspection of anatomical images and pRF centers (S3 Fig).

4.6 Predicting fMRI signals directly from ECoG models

The relationship between fMRI and ECoG signals was analyzed using a linear regression model. The cross-validated coefficient of determination (R²) was used as a metric for model accuracy, and the regression coefficients were used to test whether ECoG predictors (broadband, gamma, and alpha) had a positive or negative relation with BOLD.

The relationship between fMRI and ECoG signals was analyzed using a linear regression model: where y is a vector of fMRI amplitudes (beta estimates), with n entries for the n different stimuli; X is a matrix of ECoG responses, n by 1, 2, or 3, where the columns correspond to one or more of broadband, gamma, and alpha estimates; b are the 1, 2, or 3 beta weights for the broadband, gamma, and alpha estimates; c is a constant (the y-intercept); and ε is the residual error term. The model was fitted separately for each cortical site (electrode/ROI pair) and for different combinations of predictors—broadband alone, gamma alone, alpha alone, each pairwise combination, and all three predictors together.

The n stimuli in the regressions included the contrast patterns and the blank stimulus. Inclusion of the blank stimulus is important for capturing the sign of the mean response. For example, if all contrast patterns induced a BOLD response of a particular level (say, +1) and induced ECoG responses of a particular level (say, –1) and we did not include the blank stimulus in the regression, then after subtracting the mean from each measure, all beta estimates would be approximately 0. This would mask a systematic relationship between ECoG and BOLD measures (in this example, an anticorrelation) arising from viewing stimuli with contrast compared to viewing a blank screen.

Models were evaluated by split-half cross-validation. First, the regression model y_i = X_ib_i + c_i + ε was fit using half of the fMRI subjects (1 and 2) and half of the ECoG stimulus repetitions (even repetitions). To cross-validate this model, the beta values (b_i) were then applied to the left out half of the ECoG data (odd stimulus repetitions) to predict the left out half of the fMRI data (fMRI subjects 3 and 4). The same procedure was applied by reversing the training and testing data. This resulted in two testing datasets with BOLD responses predicted from ECoG for each stimulus condition (X_ib_i + c_i) and an actual measured BOLD value. For each cortical site, the coefficient of determination (see below) was calculated between the concatenated predictions and BOLD data values of the two test sets. All R² values reported in the results are cross-validated in this manner. The same pattern of results was achieved if instead of cross-validation, we solved the models on the complete datasets and computed the R² adjusted for the number of regressors.

To test whether different ECoG predictors (broadband, narrowband, alpha) had a positive or negative relation with BOLD, we tested whether the regression coefficient was significantly larger or smaller than 0. The regression coefficient was considered to be significantly different from 0 using a bootstrap statistic: for each model, the median of the beta values across sites was calculated after resampling 10,000 times. If <2.5% of the resampled statistics were smaller than zero, the beta values were considered significantly positive, and similarly, if <2.5% of the resampled statistics were greater than 0, the beta values were considered significantly negative.

4.7 Model accuracy

All model predictions were quantified using the coefficient of determination on cross-validated predictions. For predicting BOLD data from simulations of population neuronal activity (Fig 7, S7 Fig), the predicted BOLD has arbitrary units. In these cases, the observed BOLD and the predicted BOLD were both normalized by subtracting the mean and then dividing by the vector length. When predicting BOLD responses from features of the LFP data (broadband, gamma, and alpha) by regression, the predicted BOLD data were in the same units as the measured BOLD, and no normalizing or rescaling was done.

To quantify the accuracy of the models, we calculated the cross-validated coefficient of determination, R²: where y are the data values and f are the prediction values. Because the model fits are cross-validated, it is possible for the model errors (residuals) to be larger than the data values, hence R² can be lower than 0, and spans (−∞,1]. In the case in which the model predictions and the data are unrelated and each are normally distributed with equal variance, R² will tend to –1.