Saddle-node bifurcation and the θθθ -model

A useful hint for an appropriate parameterization of a phase model can be obtained by analyzing the type of bifurcation from the fixed point (resting potential) to oscillation (repetitive firing). As discussed in the previous chapter, neurons can be grouped into two classes according to their type of bifurcation. Class-I neurons start to oscillate via a saddle-node bifurcation, while class-II neurons go through a Hopf bifurcation.

In class-I neurons right at the bifurcation, where the node loses stability, the trajecto-ries are attracted by the homoclinic orbit of the saddle. Above the bifurcation this orbit turns into a stable limit cycle like the one shown in Fig. 3.9A. Approximating the flow field at its saddle to second order in the direction of the homoclinic orbit, results in the normal form

˙

x a I I_th bx²; a b (3.19)

3.3 SADDLE-NODE BIFURCATION AND THEθ-MODEL 45

Figure 3.14: SQUARE-ROOT f -I-CURVES. The f -I-curves simulated by two class-I conductance-based models compared with the square-root relation as expected from the saddle-node bifurca-tion.A & B f -I-curve of the Traub-Miles model.AThe overall dependence of the firing frequency on the input current I is well described by the square-root relation (3.21) of theθ-model.BFitting the square root for currents I 05 A cm²close to the threshold I_th 01 A cm²only, indeed results in an excellent approximation in this region, but fails to describe the f -I-curve for higher currents. C & D f -I-curve of the Connor model. COverall description of the whole f -I-curve.

DFocusing on the very vicinity I 84 A cm² of the threshold I_th 81 A cm², results in a perfect match of the square-root relation (3.21) to the f -I-curve. For higher currents, however, the

f -I-curve increases faster and more linearly than the square root.

of a saddle-node bifurcation (Strogatz, 1994; Hoppensteadt & Izhikevich, 1997). The input current I plays the role of the bifurcation parameter. I_th is the threshold of the f -I-curve were the saddle-node bifurcation occurs. Ermentrout (1996) applied this idea to class-I neurons. After a variable transformation he arrived at the so calledθ-model¹

τθ˙ 1 cosθ 1 cosθ c I I_th ; π θ π

θ π ; θ π spike (3.20)

whereτand c are two parameters .

The θ-model is another simple phase oscillator as a model of a spiking neuron. In contrast to the non-leaky and the leaky phase oscillator (3.2) and (3.14), it is based on the saddle-node bifurcation of class-I neurons and therefore has high predictive power. For example, the f -I-curve of theθ-model is the square-root of the input current I relative to threshold I_th:

1To describe the model dynamics in real time instead of slow time, a slightly different parameterization of theθ-model has been used.

I I_th

Figure 3.15: PROPERTIES OF THE MODIFIED θ-MODEL. ADependence of the phase velocity ˙θ on the phase angleθfor different input firing frequencies as indicated. The time constantτis set to 5 ms. At a firing frequency of f 0 Hz the graph of the phase velocity touches theθ-axis at θ 0. For even lower input currents I I_th the original model (3.20) exhibits two fixed points.

A stable one at negative phase angles (filled circle) and an unstable fixed point at positive phase angles (open circle). Bringing the system above the unstable fixed point leads to a spike before it settles at the stable fixed point. For inputs above threshold ( f 0) the phase-velocity curve is always above zero. No fixed points exist and spikes are generated periodically. BGraphs of the phase velocity for f 100 Hz and different values of the time constantτas indicated.

This relation describes many f -I-curves of both conductance-based models and real neu-rons fairly well. This square-root dependence is compared in Fig. 3.14 with the f -I-curves simulated with the Traub-Miles model (panelA & B) and the Connor model (panelC &

D). Indeed, close to the bifurcation the square-root relation (3.21) matches perfectly the f -I-curves (B & D). The overall dependence of the firing frequency on the input current can be approximately described by (3.21), too, but with different parameter values (A &

C). This inconsistency is a hint that theθ-model is valid only in the very vicinity of the bifurcation.

Deviations of the square-root relation from the real f -I-curves decrease the perfor-mance of the θ-model. To account for these differences, (3.21) can be solved for I Ith

and inserted into (3.20):

τθ˙ 1 cosθ 1 cosθ πτf I ² ; π θ π

θ π ; θ π spike (3.22)

In this version theθ-model reproduces exactly the neuron’s f -I-curve for constant stimuli I. The dependence of the phase velocity ˙θ on phase is symmetric with respect toθ 0.

The relation of the time constant τto the firing frequency f I determines whether the phase velocity during the spikes is faster or slower than in between the spikes. For f I 0 theθ-model has a single fixed point atθ 0. The properties of this phase oscillator are illustrated in Fig. 3.15.

3.3.1 Performance of the

θθθ

The performance of the θ-model for time-dependent stimuli is tested in Fig. 3.16. The θ-model predicts the spikes of the Traub-Miles model with high accuracy even for very strong and fast fluctuating stimuli. However, it predicts the spikes of the Connor model

3.3 SADDLE-NODE BIFURCATION AND THEθ-MODEL 47

standard deviation A cm²

2

standard deviation A cm²

A B

standard deviation A cm²

model #2:τ 10 ms

Figure 3.16: PERFORMANCE OF THE θ-MODEL. AThe performance tested on the Traub-Miles model is greatly enhanced compared to the performance of the leaky phase oscillator shown in Fig. 3.12A. The worst performance at the stimulus with standard deviation σ 8 A cm² and cut-off frequency fc 16 kHz is still better than 80 %. B In contrast, the performance tested on the Connor model is worse than the one of the leaky phase oscillator shown in Fig. 3.12 B, and similar to the performance of the non-leaky phase oscillator in Fig. 3.12B.C, D & EWithin the 05 ms window two θ-models with different time constantsτas indicated predict the same spikes. However, the predicted spikes are not exactly identical. In the plots the averaged distances of corresponding spikes are displayed. CThe averaged distances in milliseconds resulting from testing oneθ-model withτ 1 ms against aθ-model withτ 100 ms. The performance was tested with Gaussian white-noise stimuli with standard deviations and cut-off frequencies as indicated, and mean µ 10 A cm². The f -I-curve of the Traub-Miles model was used for the models. D A histogram of the averaged distances of the spikes fromC. Most differences are below 0.02 ms.

EA histogram of spike distances from twoθ-models with time constantsτ 1 ms andτ 10 ms.

The same stimuli as inCwere used.

worse than the leaky phase oscillator and with similar fidelity as the non-leaky phase oscillator (compare with Fig. 3.6 and Fig. 3.12).

Interestingly the performance of the θ-model is almost independent of the time con-stant τ. To further check this, two θ-models, which differed only by their values of the time constantτ, were tested against each other in Fig. 3.16C & D. Indeed, they produce very similar timing of spikes within 05 ms for the entire range of stimuli used, regard-less of their time constant ofτ 1, 10, or 100 ms. The reason for this is the independence of the response function z t of the time constant (see below). However, the spikes are not identical. On average they differ by about less than 0.01 ms.