Fig. 2.

Balanced and unbalanced solutions. (A) Perceptron solutions as a function of load and fraction of excitatory weights. Above the capacity line [${\alpha }_{\mathrm{c}}\left(f_{\mathrm{exc}}\right)$, solid line] no solution exists. Balanced solutions exist only below the balanced capacity line [${\alpha }_{\mathrm{b}}\left(f_{\mathrm{exc}}\right)$, dashed shaded line]. Between the balanced capacity and maximum capacity lines, only unbalanced solutions exist (U). On the other hand, below the balanced capacity line, unbalanced solutions coexist with balanced ones (B+U). (B) The norm of the synaptic weight vector of typical solutions as a function the load [in units of $\left(V_{\mathrm{th}}-V_{\mathrm{rest}}\right)/{\sigma }_{\mathrm{exc}}$]. Below ${\alpha }_{\mathrm{b}}$ the norm is clipped at its upper bound $\mathrm{\Gamma }$ (in this case $\mathrm{\Gamma }=\mathrm{\hspace{0.17em}1}$). Above ${\alpha }_{\mathrm{b}}$ the norm collapses and is of order $1/\sqrt{N}$ (shown here for $N=\mathrm{\hspace{0.17em}3}\text{,}000$). (C) The input imbalance index (IB, Eq. 3) of typical solutions as a function of the load. Note the sharp onset of imbalance above ${\alpha }_{\mathrm{b}}$. In B and C $f_{\mathrm{exc}}=\mathrm{\hspace{0.17em}0.8}$, yielding ${\alpha }_{\mathrm{c}}=\mathrm{\hspace{0.17em}1}$. See SI Materials and Methods for other parameters used. For simulation results see Fig. S1.