These reports thus represent upper and lower bounds on the set of well-fit activation parameters. The fourth term is a regularization term that penalizes excessively strong weights. Similar goodness-of-fits and circuit connectivities were obtained when, instead of the soft constraint described by the fourth term, we applied a fixed maximum weight Wmax = 0.1 nA. The

cost function described above consists of a sum of quadratic terms, which allowed the BVD-523 order weights onto each neuron to be fit with a constrained linear regression algorithm. Because each neuron could be fit separately from every other, the overall fitting procedure represented a sequence of 100 constrained linear regressions for 101 coefficients Wij and Ti (of which 50 are constrained to be zero, see Figure 2B). Coefficients of the different regression terms (ρinh, learn more ρexc, λ) were chosen to maximize the number of circuits that provided good fits to both the tuning curve data and the inactivation experiments (Supplemental Experimental Procedures). However, the region of well-fit activation curves and basic themes of circuit organization were not observed to change significantly over a broad range of coefficient values around the optimum. The sensitivity of the circuit to changing patterns of synaptic connectivity was calculated from the Hessian matrix Hij(k)=∂2εk/∂Wki∂Wkj described in Results. For the

individual connection weight analyses, the Hessian matrix for a given neuron (e.g., the kth lowest-threshold neuron in each circuit) was averaged across 100 circuits generated by randomly drawing tuning curves from the experimental distribution of Figure 2A (inset). Tolerance bars were generated for each connection weight onto neuron k by determining from the Hessian the amount this weight would be required to change in order to produce a noticeable (5 pA) change in the cost function. These bars then were overlaid

upon the weighted average of the optimal Levetiracetam connection weights for the 100 circuit simulations, where each model’s connection strengths were weighted by their sensitivities. Eigenvectors and eigenvalues were found for each of the 100 randomly generated circuits. To identify salient features present across circuits, we then generated the average first, second, and third, etc. eigenvectors across all 100 circuits (Figure 6E, green lines). Perturbations in Figures 6F and 6G corresponded to changing weights by a fixed vector length along all of the shown eigenvectors; thus, differences between sensitive and insensitive perturbations reflected summing (for sensitive) or cancelling (for insensitive) effects of individual weight changes, and not different sizes of weight perturbations. To produce the nth column of Figure 6G, each neuron was perturbed along its nth eigenvector.