Three- and two-dimensional bifurcation diagrams in the weak-inhibition regime.

<p>Left, <i>μ</i>_<i>E</i> is shown on the <i>z</i>−axis as function of <i>I</i>_<i>E</i>−<i>I</i>_<i>I</i>. Here we plot only the local bifurcations (blue: saddle-node curves, red: Andronov-Hopf curves) that bound the plain/dashed regions representing the stable/unstable equilibrium point areas. Right, complete set of codimension two bifurcations. The Andronov-Hopf bifurcation curves (red lines) are divided into supercritical (plain) and subcritical (dashed) portions. The supercritical/subcritical portions are bounded by a generalized Hopf bifurcation, GH, and Bogdanov-Takens bifurcations, BT. The latter are the contact points among saddle-node bifurcation curves (blue lines), Andronov-Hopf bifurcation curves (red lines), and homoclinic bifurcation curves (hyperbolic-saddle/saddle-node homoclinic bifurcations are described by plain/dashed orange curves). SNIC bifurcations identify the contact point between the saddle-node curve and the homoclinic one. From GH originate two limit point of cycles curves (dark green lines) that collapse into the homoclinic curves. Before this, they present a cusp bifurcation, CPC. Each saddle-node curve shows, in addition to BT, a cusp bifurcation, CP.</p

Codimension one bifurcation diagrams for <i>μ</i>_<i>E</i> (left) and <i>μ</i>_<i>I</i> (right) as a function of <i>I</i>_<i>E</i>, for <i>J</i>_<i>II</i> = −34 and <i>I</i>_<i>I</i> = −10.

<p>In the top panels, the stable/unstable primary equilibrium curve is described by plain/dashed black curves, while the secondary ones are described by plain/dashed violet curves. H bifurcations appear on both the primary (red) and secondary (purple) equilibrium curves, giving rise to unstable and stable limit cycles respectively (the maxima and minima of the oscillations are described by dashed and plain brown curves respectively). In particular, the unstable cycles collapse into a homoclinic bifurcation, described by an orange line. The shaded colored boxes emphasize sampled areas of the codimension one bifurcation diagram, whose corresponding temporal dynamics is shown in the bottom-panels. In particular, the figure shows the correspondence between the BP, H and LP bifurcations and the split of the inhibitory membrane potentials, the emergence of oscillations and the sudden jump of the neural activity, respectively. We induced transitions between these different kinds of dynamics by increasing linearly the external input current <i>I</i>_<i>E</i> while keeping <i>I</i>_<i>I</i> fixed.</p

Structure of the analyzed network.

<p>Neurons are fully connected and grouped into one excitatory and one inhibitory population. <i>J</i>_<i>αβ</i> represents the set of synaptic weights from the population <i>β</i> to the population <i>α</i>, while <i>I</i>_<i>α</i> is the external current (i.e. the stimulus) to the population <i>α</i>.</p

Codimension one bifurcation diagrams for <i>μ</i>_<i>E</i> (left) and <i>μ</i>_<i>I</i> (right) as a function of <i>I</i>_<i>E</i>, for <i>J</i>_<i>II</i> = −100 and <i>I</i>_<i>I</i> = −10.

<p>The colored curves in the top panels describe the same bifurcations as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.g008" target="_blank">Fig 8</a>. Besides, we observe LP bifurcations on the secondary branches (light blue points), an LPC bifurcation (dark green loop), and a TR bifurcation (gray loop). To conclude, similarly to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.g008" target="_blank">Fig 8</a>, in the bottom panels we plot some examples of dynamics of the membrane potentials for linearly increasing values of the current <i>I</i>_<i>E</i>.</p

An example of catastrophe manifold.

<p>On the left, we show an example of hysteresis displayed by the system. The plain lines describe stable equilibria, while the dashed line the unstable ones. On the right, we show an example of catastrophe manifold. The panel highlights three different behaviors of the network for increasing values of ∣<i>I</i>_<i>I</i>∣: leaky integrator, perfect integrator and switch (red curves). In particular, the perfect integrator corresponds to a cusp bifurcation (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.g007" target="_blank">Fig 7</a>).</p

Codimension two bifurcation diagram on the <i>I</i>_<i>E</i>−<i>I</i>_<i>I</i> plane for <i>J</i>_<i>II</i> = −34 (left) and <i>J</i>_<i>II</i> = −100 (right).

<p>In addition to the bifurcations already displayed in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.g007" target="_blank">Fig 7</a> (right) we stress the presence of new ones. The branching points form two curves (light green dot-dashed lines) that define the values of <i>I</i>_<i>E</i>−<i>I</i>_<i>I</i> that bound the secondary branches of equilibrium points (see the violet curves in Figs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.g008" target="_blank">8</a>) and (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.g009" target="_blank">9</a>)). The bifurcations originated on the secondary branches are differentiated from those originated on the primary one. Specifically, we show H and LP curves (purple and light blue lines, respectively). In addition, we display the torus bifurcation curves (gray lines).</p

Spontaneous symmetry-breaking under strong inhibition.

<p>When <i>λ</i>_<i>I</i> (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e029" target="_blank">Eq (8)</a>) is negative, the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e018" target="_blank">system (3)</a> has only one stationary solution for a fixed set of parameters. This solution is stable and symmetric, but if a network parameter (e.g. <i>I</i>_<i>E</i>) is allowed to vary continuously, <i>λ</i>_<i>I</i> may change sign. If this occurs, the solution becomes unstable and the network chooses randomly between two new alternative states, breaking the symmetry. The new states can be stable (e.g. for <i>N</i>_<i>I</i> = 2) or unstable. In the former case the phenomenon of spontaneous symmetry-breaking may be understood intuitively as a ball that rolls in a double well potential and reaches a state of minimum energy, which corresponds to a stable stationary solution of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e018" target="_blank">Eq (3)</a>. In the weak-inhibition regime <i>λ</i>_<i>I</i> is always negative, therefore strong inhibition is a necessary condition for spontaneous symmetry-breaking.</p

Codimension one bifurcation diagrams for <i>N</i>_<i>I</i> = 4.

<p>The left panel shows the diagram of the excitatory neurons, the central panel that of three inhibitory neurons, while the right panel shows the diagram of the remaining neuron in the inhibitory population. The secondary branches are unstable near the BP bifurcations, see text. Compared to the case <i>N</i>_<i>I</i> = 2, now the inhibitory neurons have not only different membrane potentials, but also different codimension one bifurcation diagrams.</p

Formation of multiple stationary solutions in the weak-inhibition regime.

<p>The two top panels show the nullclines of the network for fixed values of <i>I</i>_{<i>E</i>,<i>I</i>} in a weak-inhibition regime, obtained for <i>J</i>_<i>II</i> = −10 and the values of the parameters in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.t001" target="_blank">Table 1</a>. Their intersection points (black dots) correspond to the solutions of the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e024" target="_blank">system (6)</a>. The top-left panel was obtained for <i>I</i>_<i>E</i> = 10, <i>I</i>_<i>I</i> = −10, while the top-right panel for <i>I</i>_<i>E</i> = 13, <i>I</i>_<i>I</i> = −10. The <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e024" target="_blank">system (6)</a> admits multiple solutions for specific values of the parameters. The two figures at the bottom show the solutions <i>μ</i>_<i>E</i> and <i>μ</i>_<i>I</i> (bottom-left and bottom-right panel, respectively) of the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e024" target="_blank">system (6)</a> for the same values of the parameters, but with varying current <i>I</i>_<i>E</i>. The black curve represents the primary branch of the network equations, and for <i>I</i>_<i>E</i> = 10 and <i>I</i>_<i>E</i> = 13 it admits one and three solutions respectively (see the black dots at the intersection with the vertical dashed lines). The (<i>μ</i>_<i>E</i>,<i>μ</i>_<i>I</i>) coordinates of these solutions correspond to those of the black dots in the top panels of the figure. The stability of these solutions is examined for the sake of completeness in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.s006" target="_blank">S5 Fig</a>.</p

Relation between any pair of inhibitory membrane potentials in the network.

<p>This figure is obtained for <i>J</i>_<i>II</i> = −100 by plotting the four solutions [<i>μ</i>_{<i>I</i>,<i>j</i>}]_0,1,2,3 of Eqs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e040" target="_blank">14</a>) + (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e041" target="_blank">15</a>) (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.s001" target="_blank">S1 Text</a> for their analytical calculation) as a function of <i>μ</i>_{<i>I</i>,<i>i</i>}, and proves the formation of three branches of solutions of the stationary membrane potentials at the branching-point bifurcations. For example, in the case <i>N</i>_<i>I</i> = 2 (left panel), the bisector of the first and third quadrants <i>μ</i>_{<i>I</i>,<i>j</i>} = <i>μ</i>_{<i>I</i>,<i>i</i>} represents the primary branch of the network, while the other two solutions that bifurcate from the branching points represent the secondary branches. The coordinates of the branching points are given by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.e075" target="_blank">Eq (23)</a>. For the sake of clarity, we do not show the stability of the solutions, which is examined in the text. Moreover, the figure shows that for increasing <i>N</i> these bifurcations disappear, as we discuss in more detail in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004992#pcbi.1004992.s001" target="_blank">S1 Text</a>.</p