19
Eigenvalues of A for standard Hodgkin-Huxley kinetics
0.4r ×
Frequency response of isopotential HH cell
0.3
0.2
0.1
∈ 0 × ×
-0.1
-0.2
-0.3
—0.4-------------'--------------'--------------'-------------'----------x
-5 -4 -3 -2 -1 0
Reλ
Figure 2.4: Left: Eigenvalues of the 4-by-4 matrix A given in (2.13). Right: Frequency
response of the active (solid) and quasi-active (dashed) models using standard HH kinetics.
evaluated numerically via a high-order quadrature scheme such as Clcnshaw-Curtis
quadrature, as described in (Trefethen, 2007).
Not only does the eigenvalue decomposition yield an analytic solution, but also the
eigenvalues of A will tell us whether resonant frequencies exist and, if so, where in the
spectrum to look for them. Recall that if A is an eigenvalue of A and if Imag(A) ≠ 0,
then the resonant frequency is in the neighborhood of
2πlmag(λ)
Cl7t∙ — ` .
1000
where the 1000 in the denominator is to convert from milliseconds to seconds (i.e.,
from mHz to Hz). If Real(A) — 0 then ωr is exactly the resonant frequency. However,
the quasi-active system is dissipative, meaning that it is possible that some eigenvalues
have Real(A) ≠ 0, and thus the above equation is only an approximation to ωr.
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