Biomedical Engineering Reference
In-Depth Information
it is not uniform. Using on the eigenslope method, the present study is
the rst to show that the simplifying assumption of uniform capacitance in
neurons has not appreciably aected the ndings in previous studies. So,
our present understanding of how brain cells propagate signals and operate
in health and in neurological and psychiatric conditions requires no mod-
ication at present due to the recent discovery of non uniformities in the
cellular distribution of membrane capacitance.
7. Conclusion
The question is often asked of why analytic theory is important when dif-
ferential equations can be solved numerically relatively quickly with com-
putational methods. The answer is that mathematical analytics provides
understanding of the underlying structure of the solution, and not just a
set of numbers called the numerical solution. Understanding ows most
naturally from simple geometrical insights. The eigenslope is such a case
of a simple geometric relation between eigenvalues and Fourier coecients
within each eigenfunction.
The eigenslope method (1) simplies the concept of Fourier coecients
and eigenfunctions, (2) provides a new method of solving for Fourier coe-
cients analytically based simply on dierentiating the eigenvalue function,
and solving for the slope at each eigenvalue, (3) enables manual graphi-
cal or numerical determination of Fourier coecients by the Runge-Kutta
method
51
without rst having the analytic solution, and (4) provides an
easy, independent method of validating analytical solutions numerically.
The eigenslope method was used derive the solutions for an analytic
mathematical model of membrane cylinders (such as tube- or ber-shaped
cells or cylindrical processes such as dendrites) with exponentially-varying
membrane capacitance. Solved models are provided for point, stepped, and
exponential changes in capacitance with distance along a membrane cylin-
der. Comparison of the passive voltage responses of the three models to
impulse stimuli and curve tting of the responses to experimental voltage
responses curve data from neurons led to the exponential model to be the
selected model. Variation in the membrane capacitance of the exponential
model of 5%, which is the range of capacitance variation found experi-
mentally by others, produced only a 1.5% change in the half-time of the
responses to impulses.
The widespread assumption of uniform membrane capacitance over the
surface of a cells is thus a valid approximation, in the sense that the degree
Search WWH ::
Custom Search