Dynamical Systems in Neuroscience:

80 One-Dimensional Systems20 mV100 ms-60 mVslow transition0 pA43.1 pAFigure 3.29: A 400-ms latency in a layer 5 pyramidal neuron of rat visual cortex.-40unstable equilibriamembrane potential, V (mV)-45-50stable equilibriasaddle-node(fold) bifurcation-550 5 10 15 20 25injected dc-current I, (pA)Figure 3.30: Bifurcation diagram of the system in Fig. 3.26.ability of many neocortical neurons, such as the one in Fig. 3.29, to generate repetitiveaction potentials with small frequency, and how it predicts that all such neurons,considered as dynamical systems, reside near saddle-node bifurcations.3.3.6 Bifurcation diagramThe final step in the geometrical bifurcation analysis of one-dimensional systems isthe analysis of bifurcation diagrams, which we do in Fig. 3.30 for the saddle-nodebifurcation shown in Fig. 3.26. To draw the bifurcation diagram, we determine thelocations of the stable and unstable equilibria for each value of the parameter I andplot them as white or black circles in the (I, V ) plane in Fig. 3.30. The equilibria formtwo branches that join at the fold point corresponding to the saddle-node bifurcation(hence the alternative name of fold bifurcation). The branch corresponding to theunstable equilibria is dashed to stress its instability. As the bifurcation parameter Ivaries from left to right passing through the bifurcation point, the stable and unstable