Fig 1.
Summary of SBI workflow used to infer model parameters that can account for recorded neural dynamics. 1) A prior distribution of assumed relevant model parameters and ranges is constructed. 2) A dataset of simulated neural activity patterns is generated with parameters sampled from the prior distribution. 3) User defined summary statistics are chosen to describe waveform features of interest. 4) A specialized deep learning architecture is trained to learn the mapping from neural activity constrained by summary statistics to underlying model parameters. 5) Specific neural activity patterns of interest are fed into the trained neural network, which subsequently outputs a distribution over the potential underlying model parameters. 6) Parameter estimates for different waveforms can be compared through diagnostics like parameter recovery (if the ground truth is known), or posterior predictive checks.
Fig 2.
A: Local network connections of the HNN model include: 1) excitatory AMPA and NMDA synaptic connections (black circles) originating from pyramidal neurons (blue), and 2) inhibitory GABAA and GABAB synaptic connections (black lines) originating from inhibitory interneurons (yellow). B: Proximal exogenous input connection pattern. C: Distal exogenous input connection pattern, see text for further description. D: 3D rendering of full neocortical column model. Figure adapted from Neymotin et al. 2016 [5].
Fig 3.
A: Simulated voltage and current dipole waveforms are shown for two exemplar parameter configurations with latency between positive I+ and negative I− square current injections, Δt = 0 (blue), and Δt ≠ 0 (orange). An example of the original “Raw” waveform (top), as well as the PCA transformed waveform (bottom) with the first 30 components (PCA30) are shown to demonstrate that this summary statistic retains almost identical information. B: Posterior distributions showing the inferred values that can generate the blue and orange waveforms from panel A (PCA30 used to generate distributions) demonstrate that when the latency Δt between the inputs is zero, their amplitudes are indeterminate as visible as a highly dispersed distribution on panels B(a-c, blue), and with a positive correlation between the parameters I+ and I− on panel B(b, blue). In contrast, when Δt ≠ 0 (orange), the distributions are tightly concentrated around the ground truth parameters (stars on panels B(a,c)) used to generate each simulation. The posterior distributions for the parameter Δt are concentrated around the ground truth parameters for both conditions (panel B(f)). C: Schematic of how RC circuit is driven by positive I+ and negative I− square current injections, where the amplitude and latency Δt between pulses serve as parameters. This simulation parallels HNN simulations below in which a single excitatory proximal/distal input with variable synaptic conductances and latencies produce positive/negative deflections in the current dipole.
Table 1.
Simulation parameters and SBI training.
Fig 4.
Diagnostics to compare summary statistics on RC circuit model.
A: Summary statistics applied to the simulated time series included: PCA30 (i), PCA4 (ii), Peak (magnitude and timing of max/min, iii), and BandPower (four bands between dotted lines, iv). PCA plots (i-ii) show the associated inverse transformed signal. Two exemplar simulations with pulse latencies Δt = 0 (blue) and Δt ≠ 0 (orange) are shown. B: Conditioning the approximate posterior distribution on the Δt = 0 time series produces indeterminacies for all summary statistics, such that the ground truth (red dotted lines) current injection amplitudes (I+ and I−) cannot be uniquely recovered. PCA30, PCA4, and Peak features exhibit a linear interaction between parameters for the Δt = 0 time series, whereas the ground truth is recovered for the Δt ≠ 0 (orange) time series. BandPower produces non-linear interactions for both time series. C: Local parameter recovery error (PRE) heatmaps are shown. Brighter colors indicate higher dispersion of the posterior around the ground truth parameters defined by each square. Errors tend to be concentrated around Δt = 0 for PCA30, PCA4, and Peak features. D: Local posterior predictive check (PPC) heatmaps are shown. Brighter colors indicate regions where simulations generated from posterior samples are further from the conditioning time series. For both diagnostics, it is clear that PCA30 produces the PRE and PPC across the parameter grid. The ground truths of the exemplar simulations of panels A/B are indicated by blue/orange squares.
Fig 5.
HNN simulations that mimic RC circuit.
HNN simulations that reflect the nearly identical parameter configuration as the RC circuit in Fig 3. A: Simulated current dipole waveforms are shown for two exemplar parameter configurations with Δt = 0 (blue) and Δt ≠ 0 (orange). The original “Raw” simulated waveform (top) is plotted in comparison with the PCA inverse transformed waveform with 30 components (PCA30, bottom). B: Posterior distributions showing the inferred values that can generate the waveforms from panel A demonstrate that when the latency between the inputs is zero (blue), their amplitudes are indeterminate as visible as a dispersed distribution on panels B(a-c, blue), with a positive correlation between the parameters P and D on panel B(b, blue). Unlike the previous example (Fig 3), the indeterminacy is notably smaller for Δt = 0, with the posterior distributions primarily being concentrated around the ground truth parameters for P and D (stars on panels B(a,c)). C: Schematic of HNN simulations in which a single excitatory proximal/distal input with variable synaptic conductances and latencies produce positive (red)/negative (green) deflections in the current dipole.
Fig 6.
SBI diagnostics of summary statistics in HNN.
The analysis shown in Fig 4 is repeated on a simplified HNN simulations for comparison. A: Summary statistics included PCA30 (i), PCA4 (ii), Peak (iii), and BandPower (iv). Two exemplar simulations with input latencies Δt = 0 (blue) and Δt ≠ 0 (orange) are shown. B: We show the approximate posterior when conditioned on both the exemplar waveforms. The Δt = 0 time series produces a positive correlation between P and D for all summary statistics. C: Local parameter recovery error (PRE) is shown. Unlike the RC circuit, PCA30 and PCA4 permit better ground truth recovery even when Δt is near zero. In contrast, Peak features have poor parameter recovery similar to the RC example D: Local posterior predictive checks (PPC) are shown. PCA30 and PCA4 produce the values across the parameter grid.
Fig 7.
HNN-SBI recovers circuit mechanism of Beta Event magnitude described in previous studies.
A: Schematic of Beta Event simulations in HNN. Beta Events are generated by a simultaneous burst of subthreshold proximal (red) and distal (green) excitatory inputs to L5 pyramidal neurons. B: Average source localized MEG Beta Event waveforms recorded from two subjects. Subject 1 (top, blue) exhibits a larger magnitude trough compared to subject 2 (bottom, orange). Simulations corresponding to a posterior predictive check (PPC) are shown in black, such that the parameters were sampled from the posterior (panel C) of each respective waveform. C: Posterior distributions conditioned on large (blue) and small (orange) magnitude Beta Events demonstrate that larger proximal variance produces a larger magnitude trough. Overlap coefficients (OVL) quantifying the separability of the marginal posterior distributions conditioned on each waveform are shown on the diagonal for the corresponding parameters. The marginal distributions for distal variance are highly overlapping, and non-overlapping for the proximal variance D: PRE heatmap of shows accurate parameter recovery (dark colors) when the ground truth parameters of
are around 5 ms2, and quickly worsen (light colors) as
increases or decreases. E: PRE heatmap of
shows accurate parameter recovery across the entire range of the prior distribution for both
and
.
Fig 8.
Inferring local connectivity parameters from ERP waveforms.
A: Schematic of ERP simulations in HNN. Evoked activity is driven by a fixed sequence of proximal-distal-proximal exogenous inputs. SBI is used to infer the maximal conductance strength () of local excitatory/inhibitory connections to the proximal/distal dendrites of L5 pyramidal neurons for example waveforms. B: Exemplar simulated ERPs (blue and orange solid lines) with differing local connectivity strengths chosen from a defined prior distribution (described in the text) are shown, along with the fixed timing of the sequence of exogenous inputs for each simulation (red and green arrows). C: Spike raster plots of cell specific firing for the two ERP simulation conditions from panel B. D: Posterior distributions over local connectivity parameters alongside ground truth parameters (stars on diagonal) for conditioning observations. A strong interaction between excitatory/inhibitory distal inputs (EL2 and IL2) is visible in the lower square. Overlap coefficients (OVL) quantifying the separability of the marginal posterior distributions conditioned on each waveform are shown on the diagonal for the corresponding parameters. EL2 and IL2 exhibit a small amount of overlap with OVL values of 0.011 and 0.190 respectively. In contrast EL5 and IL5 were much more distinguishable, exhibiting OVL values of 2.28e-5 and 1.59e-13 respectively. E: Local parameter recovery error (PRE) for distal inhibition IL2 indicates errors are higher for observations generated with strong excitatory EL2 and weak inhibitory IL2 distal connections.