.
Fitting the Rapid Theta Neuron Model to the EIF
In Chapter 3 the rapid theta neuron model with variable AP onset rapidness r was introduced.
It was built to yield the membrane time constant at the stable fixed pointVS =−1/2 and an r times more rapid AP initiation at the unstable fixed pointVU =1/2. In the dimensionless voltage representation the rapid theta neuron model is defined as
τmd ¯V dt =
(aS(V¯ −V¯G)2−I¯T+I(t)¯ V ≤VG
aU(V¯−V¯G)2−I¯T +I(t)¯ V >VG, (A.7) with
V¯G = 1 2
r−1 r+1 I¯T = 1
2 r r+1 aS = r+1
2r aU = r(r+1)
2 =r2aS.
FI-curves
In this section, we investigate how the rapid theta neuron model can be fit to the exponential integrate and fire model (EIF). Originally, the EIF model was introduced by Fourcaud-Trocmé et al. [88] to fit the Wang-Buzsáki model of hippocampal interneurons [38]. It was, however, necessary to introduce a refractory period to fit the EIF to the Wang-Buzsáki model. They also fit the QIF model (r=1) with almost identical parameters but without a refrectory period to the Wang-Buzsáki model. The parameters obtained by Fourcaud-Trocmé et al. are summarized in
EIF (Fourcaud-Trocmé [88]) QIF (Fourcaud-Trocmé [88]) EIF (Badel [126])
τm 10 ms 10 ms 17.2 ms
gL 0.1 mS/cm2 0.1 mS/cm2
-VL −65 mV - −57.0 mV
VT −59.9 mV −59.9 mV −42.0 mV
4T 3.48 mV 3.48 mV 1.51 mV
τr 1.7 ms - 10 ms
VR −68 mV −63.8 mV −55 mV
Table A.1 –Model parameters of the exponential integrate and fire (EIF) model and the quadratic integrate and fire (QIF) model given in Ref. [88, 126]. The EIF and QIF models in the second and third column are fit to the Wang-Buzsáki model. The EIF model in the fourth column is fit to a measureIV-curve of a cortical neuron (see Fig. A.3).
0 1 2 3 4 5
I (µA/cm2) 0
100 200 300
ν (Hz)
EIF with τr = 0 EIF
QIF VR = -63.8mV Theta
0 0.05 0.1 0.15 0.2
I (µA/cm2) 0
5 10 15 20
ν (Hz) r = 2
Figure A.2–Tonic firing rates versus suprathreshold input current (FI-curves) of different integrate and fire models. The EIF model with refractory period and the QIF model with finite reset are fit to the Wang-Buzsáki model in Ref. [88] (parameters in Table A.1). Additionally is shown the EIF model without refractory period and the theta neuron model and rapid theta neuron model with AP onset rapidnessr=2.
Table A.1. Because the QIF model had a finite reset, it does not perfectly correspond to the theta neuron model used in Chapter 2. In fact, these subthreshold properties, such as refractory period and reset voltage strongly influence theFI-curve, to which the models were fit. InFigure A.2are compared the models from Ref. [88] with two additionalFI-curves of the EIF model without the refractory period and the QIF model with reset to −∞equivalent to the theta neuron model. We have also displayed theFI-curve of a rapid theta neuron with AP onset rapidnessr=2. It can be seen, that the FI-curves of the different models and with different parameters differ considerably and it is questionable to use theFI-curve to obtain a good fit. The authors in Ref. [88] have used other measures for a comparison of the ability of a neuron to track fast changes in the input and came to the conclusion that the speed with which neurons can track changes in inputs is related to the spike slope factor determining the AP onset rapidness. These methods are, however, beyond the scope of this thesis but were applied to a similar neuron model in Ref. [127].
Another possibility for a fit is to look at the onset of the FI-curve. This is characteristic of the bifurcation that the neuron undergoes at the threshold to tonic firing. For type I excitable neurons this onset is ν ≈β√
I−IT. The parameterβ for the EIF model is given in Ref. [88] and confirmed by the data in Fig. A.2 toβ=0.038 ms−1µA−1/2cm. For the rapid theta neuron model,
Figure A.3–Experimentally obtained membrane current versus membrane potential (IV-curve) of a cortical neuron and the fit to the EIF model from Ref. [126].The parameters of the exponential integrate and fire (EIF) model are given in Table A.1.
the interspike interval, Eq. (3.11), and the conversion to the voltage representation, Eq. (A.2), yield
β = 1
π τm√ 4TgL
r r r+1. This solved for the AP onset rapidnessryields
r
r+1 = 4TgLπ2τm2β2 r = 4TgLπ2τm2β2
1− 4TgLπ2τm2β2.
The parameters yieldr=1, not too surprisingly because the EIF model and the QIF model share the same parameters in Ref. [88]. But we can check this with another EIF model that was fit to the experimentally obtainedIV-curve of a cortical neuron.
Badel et al. measure the membrane current of a real cortical neuron and fit the parameters of the EIF model to match the data [126]. Their parameters are given in Table A.1. Because they use the third representation obtained by (A.5), we get
β = 1
π√
24Tτm
r r r+1. This solved for the AP onset rapidnessryields
r = 4Tτmπ2β2 1− 4Tτmπ2β2.
0 0.05 0.1 0.15 0.2 I (mV/ms)
0 5 10 15 20
ν (Hz)
Fourcaud Badel
0 0.005 0.01 0.015
I (mV/ms) 0
2 4 6
ν (Hz)
fit with β=0.04mV-1
Figure A.4–Tonic firing rates versus suprathreshold input current (FI-curves) of the EIF model fit to the Wang-Buzsáki model [88] and to a cortical neuron [126].The parameters of the exponential integrate and fire (EIF) model are given in Table A.1.
A numerical fit to the FI curve of the EIF model with the parameters from Ref. [126] yields β =0.040 mV−1/2ms−1/2, comparable to an AP onset rapidnessr=0.7 (Fig.(A.4)).
IV-curves
Another possibility to fit the rapid theta neuron model to the EIF models from Ref. [88] and [126]
is to use the membrane current directly. Because in the EIF model the membrane currentImgrows exponentially with the voltage V, and in the rapid theta neuron model it grows quadratic, this is only representative at one specific voltage. We therefore measured the slope of theIV-curve at the stable and unstable fixed point. With the parameters from Fourcaud-Trocmé et al. [88], we get a ratio ofr=1.9, whereas with the parameters from Badel et al. [126], we getr=12.