Fig 1.
The integrative loop between experiment, model, and theory.
Black arrows represent the classical forward modeling approach: Experimental anatomical data are integrated into a model structure, which gives rise to the activity via simulation. The model activity is compared with experimental functional data. The usual case of disagreement leads to the need to change the model definition. By experience and intuition the researcher identifies the parameters to be changed, proceeding in an iterative fashion. Once the model agrees well with a number of experimental findings, it may also generate predictions on the outcome of new experiments. Red arrows symbolize our new approach: informed by functional data, an analytical theory of the model identifies critical connections to be modified, refining the integration of data into a consistent model structure and generating predictions for anatomical studies.
Fig 2.
Activity flow in an illustrative network example.
Left column: Global stability analysis in the single-population network. A Illustration of network architecture. C Upper panel: Input-output relationship for external Poisson drive
shown in gray. In addition
for
for the noiseless case (blue) and the noisy case (red). The inset shows the gray curve over a larger input range. Lower panel:
for different rates of the external Poisson drive
from black to light gray. Intersections with the identity line (dashed) mark fixed points of the system, which are shown in E as a function of νext. F Flux
in the bistable case for
in black,
in blue, and modified system
in red. Intersections with zero (dashed line) mark fixed points. The inset shows an enlargement close to the LA fixed point. Horizontal bars at top of figure denote the size of the basin of attraction for each of the three settings. Right column: Global stability analysis in the network of two mutually coupled excitatory populations. B Illustration of network architecture. D Flow field and nullclines (dashed curves) for
and separatrices (solid lines), LA fixed point (rectangle), HA fixed point (cross) and unstable fixed points (circles) for
in black,
in blue, and
in red. The red separatrix and the red unstable fixed point coincide with the black ones. G Enlargement of D close to the LA fixed points. Flow field of original system shown in black, of modified system in red.
Fig 3.
A Sketch of the microcircuit model serving as a prototype for the areas of the multi-area model (figure and legend adapted from figure 1 of Potjans and Diesmann [23], with permission). B Sketch of the most common laminar patterns of cortico-cortical connectivity of the multi-area model. C Population-averaged firing rates encoded in color for a spiking network simulation of the multi-area model with low external drive . D As C but for increased external drive
. The color bar refers to both panels. Areas are ordered according to their architectural type along the horizontal axis from V1 (type 8) to TH (type 2) and populations are stacked along the vertical axis. The two missing populations 4E and 4I of area TH are marked in black and firing rates
in gray. E Histogram of population-averaged firing rates shown in C for excitatory (blue) and inhibitory (red) populations. The horizontal axis is split into linear- (left) and log-scaled (right) ranges. F as E corresponding to state shown in D.
Fig 4.
Application of the mean-field theory to the multi-area model.
A Trajectories of Eq (1) starting inside the separatrix converge to an unstable fixed point. Trajectories starting close to the separatrix are initially attracted by the unstable fixed point but then repelled in the fixed point’s unstable direction and finally converge to a stable fixed point. B Firing rate averaged across populations over time. Integration of Eq (1) leads to convergence to either the low-activity (LA) or the high-activity (HA) attractor for different choices of the external input factor , with
the original level of external drive. We show eight curves with
varying from 1.0 to 1.014 in steps of 0.002 and two additional curves for
. The curves for the largest factor
that still leads to the LA state and for the smallest factor
that leads to the HA state are marked in blue and red, respectively. The four curves with
coincide with the blue curve. C Euclidean norm of the velocity vector in the integration of Eq (1) for the different choices of
. The vertical dashed line indicates the time
of the last local minimum in the blue curve. D Stationary firing rate in the different areas and layers of the model in a low-activity state for
as predicted by the mean-field theory (same display as in Fig 3). E As D, but showing the high-activity state for
.
Fig 5.
Eigenspectrum analysis of network stability.
A Eigenvalue spectrum of the effective connectivity matrix for the first (blue squares) and second (red dots) iteration. The dashed vertical line marks the edge of stability at a real part of 1. B Contribution
(Eq (13)) of an individual eigenprojection
to the shift of the unstable fixed point versus the relative change in indegrees associated with
for the first iteration. The data point corresponding to
is marked in red. The inset shows the relative angles between
and the eigenvectors
. The red line corresponds to the critical eigendirection. C Entries of the eigenvector
associated with
in the populations of the model. The affected areas are 46 and FEF. D Same as C for the second iteration.
Fig 6.
Unstable fixed points in subsequent iterations.
Population firing rates at the unstable fixed point as predicted by the mean-field theory encoded in color for iterations 1 (A) and 2 (B). Same display as in Fig 3.
Fig 7.
Altered phase space and modified connections.
A Firing rates averaged across populations 5E and 6E and across areas for different stages from the original model (black) to iteration 4 (light gray) as a function of , predicted by the mean-field theory. B Relative changes in the indegree
between areas A, B in the first iteration. C Layer-specific relative changes
in the connections within and between areas FEF and 46, for the first iteration. Populations are ordered from 2/3E (left) to 6I (right) on the horizontal axis and from 6I (bottom) to 2/3E (top) on the vertical axis as in panel D. D Relative changes in population-specific indegrees summed over target populations,
, combined for iterations two, three and four.
Fig 8.
Analysis of changes in connectivity.
A Top panel: relative changes in population-specific intrinsic indegrees summed over target populations and averaged across areas, . Bottom panel: changes in the indegrees within and between exemplary areas V4 and CITv relative to the total indegrees of the target populations, i.e.,
. Populations ordered as in Fig 7C. B Pearson correlation coefficient of the changes of the internal indegrees
between all pairs of the 32 areas. Areas ordered according to hierarchical clustering using a farthest point algorithm [31]. The heights of the bars on top of the matrix indicate the architectural types of the areas (types 1 and 3 do not appear in the model) with color representing the respective clusters. C
of the modified connectivity after 4 iterations versus the original
of the model. Only
are shown for a better overview. The overlapping red dots represent the connections between areas 46 and FEF. Unity line shown in gray. D Histogram of the cumulative changes in
over all four iterations
.
Fig 9.
Improved low-activity fixed point of the model.
Population-averaged firing rates for encoded in color (A) predicted by the analytical theory and (B) obtained from the full simulation of the spiking network. Same display as in Fig 3. C Analytical versus simulated firing rates (black dots) and identity line (gray). D Histogram of population-averaged simulated firing rates. Same display as in Fig 3.
Table 1.
Specification of the neuron and synapse parameters.