Topologically Defined Neuronal Networks Controlled by Silicon Chips

CHAPTER 2. NETWORKS OF DEFINED TOPOGRAPHY
into A.
Iinj,A = GA(VA,∞ − V0A) + GAB(VA,∞ − VB,∞) + GAC(VA,∞ − VC,∞) (2.32)
GB(VB,∞ − V0B) = GAB(VA,∞ − VB,∞) + GBC(VC,∞ − VB,∞) (2.33)
GC(VC,∞ − V0C) = GAC(VA,∞ − VC,∞) + GBC(VB,∞ − VC,∞) (2.34)
GAB is the conductivity of the synapse between A and B. Index i denotes which neuron the current
Iinj,i is injected into. The equations characterizing the situation before the current step are almost identical,
but this time Iinj,i = 0 and index k in Vj,k is set to 0 instead of ∞.
To determine the synaptic conductivities GAB, GAC and GBC, additional data are necessary from current
step measurements with B and C being presynaptic. Providing another 12 equations, the resulting
equation system yields rather lengthy solutions.
Equations 2.32 - 2.34 can be brought into the same form as 2.21 and 2.23 by eliminating VC from 2.32
and 2.33 with 2.34 and some rearrangements. This means that the three-neuron network again behaves
like an isolated cell pair if probed with just two electrodes, one in A the other in B. Of course, the
values for the somatic conductance and resting potentials differ from GA, V0A etc. as these ‘equivalent’
quantities are governed by the network properties rather than the individual cells. Calculating the
conductance between neuron A and B in analogy to an isolated pair results in a quantity G approx
AB that is
related to GAB by:
G approx
AB
= GAB +
GACGBC
GAC + GBC + GC
(2.35)
For typical values of the somatic and synaptic conductance of snail neurons (Gsyn=0.5nS and
Gsoma=2.0nS), the relative deviation between G approx
AB and GAB can be approximated by Gsyn/Gsoma.
It is on the order of some ten percent, similar to the error introduced by using the isopotential model
instead of the detailed model, which accounts for the neurites’ cable properties.
In principle, all unknown parameters of an n neuron network with an arbitrary connection pattern can
be determined from current step measurements. Based on the n 2 + n equations describing the system,
the 2n soma parameters and the at most (n − 1)n/2 synapses (the ones in parallel are regarded as a
single synapse) can be calculated. Experimental constraints, e.g. the limited number of neurons that
can be probed simultaneously, impede this approach rapidly, however.
Alternatively, the ‘equivalent’ synaptic conductance between the embedded pair A and B in fig. 2.12D
can be calculated from eq. 2.25 for the same reasons as discussed above. The deviation from the real
value of GAB decreases with increasing length of the path parallel to the synapse directly coupling
A with B. It is about (Gsyn/Gsoma) m where m denotes the number of neurons connected in series
between A and B. For every additional path in parallel, the error increases because their conductances
add up in analogy to the situation depicted in fig. 2.12A. In any case, the value obtained from eq. 2.25
is larger than GAB.
2.4.3 Active model: Hodgkin-Huxley dynamics
Most interesting electrical properties of neurons, including their ability to fire action potentials and
propagate them actively, arise from the nonlinear behavior of their membrane conductances. These
unique features have been omitted so far to simplify the mathematical description, ohmic resistors
being used instead. In reality, membrane current is mediated by transmembrane channels that are often
highly selective to one sort of ion and whose conductivity is controlled by the membrane potential
or ligands bound to specific receptors. Ligand gated channels are essential in signal transmission via
chemical synapses, but they are hardly involved in action potential generation. This process is governed
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