Cortical Network Models of Firing Rates in the Resting and Active States Predict BOLD Responses

Abstract

Measurements of blood oxygenation level dependent (BOLD) signals have produced some surprising observations. One is that their amplitude is proportional to the entire activity in a region of interest and not just the fluctuations in this activity. Another is that during sleep and anesthesia the average BOLD correlations between regions of interest decline as the activity declines. Mechanistic explanations of these phenomena are described here using a cortical network model consisting of modules with excitatory and inhibitory neurons, taken as regions of cortical interest, each receiving excitatory inputs from outside the network, taken as subcortical driving inputs in addition to extrinsic (intermodular) connections, such as provided by associational fibers. The model shows that the standard deviation of the firing rate is proportional to the mean frequency of the firing when the extrinsic connections are decreased, so that the mean BOLD signal is proportional to both as is observed experimentally. The model also shows that if these extrinsic connections are decreased or the frequency of firing reaching the network from the subcortical driving inputs is decreased, or both decline, there is a decrease in the mean firing rate in the modules accompanied by decreases in the mean BOLD correlations between the modules, consistent with the observed changes during NREM sleep and under anesthesia. Finally, the model explains why a transient increase in the BOLD signal in a cortical area, due to a transient subcortical input, gives rises to responses throughout the cortex as observed, with these responses mediated by the extrinsic (intermodular) connections.

Fig 1. Shown are two modules (regions of interest, ROI₁ and ROI₂), represented by ellipses containing pools of excitatory neurons (red triangles) and inhibitory neurons (green circles), synaptically connected within and between pools (black arrows).

The extrinsic (associational) axons between ROI₁ and ROI₂ originate and end on single neurons and are either excitatory to excitatory (red arrows) or excitatory to inhibitory (green arrows). The subcortical inputs are also excitatory; computationally, this input is represented by a Poisson train of firings, the strength and frequency of which can be varied. (B), extrinsic (intermodular, associational) connections in an 8 modular network (NI). The bold line within each module indicates separation of excitatory and inhibitory neurons within the module with all extrinsic fibers between the modules forming excitatory synapses, primarily on excitatory neurons within the modules but not exclusively so. (C), extrinsic (intermodular, associational) connections in a 7 modular network (NII). The connectivity due to the associational axons follows the criteria described in A above. NII can be taken to represent the Default Mode Network (DMN). In this case the modules may be identified as follows: 1, right lateral temporal cortex; 2, right inferior parietal lobule; 3, left inferior parietal lobule; 4, posterior cingulate cortex; 5, dorsal medial prefrontal cortex; 6, ventral medial prefrontal cortex; 7, left lateral temporal cortex. The number and diversity of the associational fibers were chosen so as to reflect the reported weight of such connections between the modules of the DMN (see [49], their Fig 8 and associated Table; also Fig 4A and 4C in [50]. (After Fig 1 in (15) with permission.)

Fig 2. (A), correlation between the standard deviation (S.D.) of the firing rate and the mean firing rate in the modules of NI (upper panel) and NII (lower panel). These frequencies and their S.D.s were determined both for the networks given in the Fig 1B and 1C as well as for these with a variety of different extrinsic (intermodular) connection weights. (B), correlation between the amplitude of the RMS BOLD signals and the S.D. of the frequency of firing rates giving rise to the BOLD signals for NII in the resting state. The BOLD evaluations and the S.D.s of the frequencies were determined for NII as for the networks in (A); correlation coefficient 0.82. (C), correlation between the amplitude of the RMS BOLD signals and the mean frequency of firing rates giving rise to the BOLD signals for NII in the resting state. The BOLD evaluations and the firing rate frequencies were determined as for the networks in (A). The gradient of the regression line is 0.06 with correlation coefficient 0.66. (D), correlation between the experimental ‘resting state fMRI amplitude’ (BOLD amplitude) and the CMR_glc(ox) in the resting state in different brain regions of humans.

The gradient of the regression line is 0.04, with correlation coefficient 0.98. Values are from Table 1 in [3].

Fig 3. The BOLD amplitude is linearly related to both the mean firing rates and their S.D. down to frequencies in the range of those observed in the cortex of primates.

Dependence of the BOLD amplitude on the S.D. of the firing rate (A) and of the mean firing rate (B) when this was reduced by decreasing the subcortical input to the modules until the mean frequency was less than 1.25Hz while keeping the extrinsic (intermodular) connection weights constant. The linear relation between the BOLD amplitude on the one hand and both the mean firing rate (B) and the S.D. of this (A) on the other is maintained as the frequency is reduced.

Fig 4. The relation between the mean firing rate in modules of NII (Fig 1C) and the mean BOLD correlations between the modules as either the coupling strength between modules or their subcortical input frequency are changed or both are changed at the same time.

The changes in average firing rate (A), and the average BOLD correlations between modules (B), are shown for changes in the subcortical input frequency and coupling strength. The changes in the average firing rate versus changes in BOLD correlations when these are altered by changing only (C) the subcortical input frequency (25 Hz to 35 Hz) at constant connection strength (0.45) or (D) the connection strength (0.2 to 0.5) at a constant input frequency (35 Hz) are given by the graphs. The gradient in C is 54 Hz and that in D is 29 Hz (linear regression lines drawn, with in (C) y = 54x – 1.9 and in (D) y = 29x + 8). In E the changes in the average firing rate versus changes in BOLD correlations are given when these are altered by changing both the subcortical input frequency and the coupling strength simultaneously, a linear relation between these being assumed. The gradient of the line in E is 41 Hz (y = 41x + 1).

Fig 5. (A) The transient BOLD signal in each module in NI (columns numbered 1 to 8) following successive 20s increases in the subcortical input to modules 4 to 7 (rows numbered 4 to 7), causing increased firing in these modules. The input was simulated by increasing the subcortical input for 20s periods. (B) gives the same results as in (A) except that individual transients are amplified sufficiently to observe them in each of the 8 modules.

This figure should be compared with that of Fig 4C in [10] in which similar experimental transients were observed over the cortex in a simple psychological test.