75
dynamics (Г = 0.990) are reproduced with kv = kj = 30, a 46-fold reduction in
dimension. Additionally, the reduced systems show greater speed-ups versus the full
systems than do the reduced systems using the HH model. Examination of voltage
traces (Figure 3.2) shows that not only are the spike times correct, but in fact the
sub- and suprathreshold voltage dynamics agree very well.
One may wonder why the speed-up is basically one order of magnitude while the
dimension reduction is two orders of magnitude. The reason lies in the fact that the
full system matrices are large but sparse, whereas the reduced matrices are small
but dense. Hence the associated matrix-vector products have different computational
costs, and we ought not expect the same scalings.
Table 3.2: Performance of reduced model (here kυ = kf~) of HHA fiber, N = 1401, as
compared with the full model.__________________________
|
kv |
Speed-up |
% Matched |
% Mismatched |
Γ |
|
10 |
8× |
81.7 |
64.3 |
0.457 |
|
15 |
7.5× |
77.7 |
7.7 |
0.840 |
|
20 |
6.9× |
95.8 |
0.5 |
0.976 |
|
30 |
5.8× |
98.6 |
0.5 |
0.990 |
3.3.2 Challenges of Branched Cells
Now consider the forked neuron which has one mother branch and two daughter
branches, all with radius 1 μm and length 500 μm. The cell consists of N = 1501
compartments having HH kinetics. The first step in constructing a reduced model is
to generate a sufficiently rich set of snapshots, but this simple cell illustrates that the
task is not trivial.