The cost of linearization

Abstract

Linear additivity of synaptic input is a pervasive assumption in computational neuroscience, and previously Bernander et al. (Journal of Neurophysiology 72:2743–2753, 1994) point out that the sublinear additivity of a passive neuronal model can be linearized with voltage-dependent currents. Here we re-examine this perspective in light of more recent findings and issues. Based on in vivo intracellular recordings, three voltage-dependent conductances seem to be of interest for pyramidal cells of the forebrain: two of them are amplifying, I _NaP and I _h ; and one of them is attenuating, I _A . Based on particular I-V characteristics reported in the literature, each of these three voltage-dependent currents linearizes a particular range of synaptic excitation. Computational simulations use a steady-state, one-compartment model. They establish maximal linear ranges, where supralinear effects—due to adding too much of any one conductance—limit these ranges. Specific, carefully selected pairwise combinations of these currents can linearize larger ranges than either current alone. In terms of parameters, the steady-state I-V characteristics of each current are critical. On the other hand, the relationships between the results here and resting conductance to ground, synaptic conductance, and number of active synapses are simple and easily scaled; thus in regard to these three latter dependences, the results here are easily generalized. Finally, to improve our understanding of evolved function, the relative metabolic costs of linearization are quantified. In one case, there is a clear preference arising from this cost consideration (a particular I _h , I _NaP pairing is less costly compared to a particular I _A , I _NaP pairing that produces an equivalent, linearized range). However in other cases, a preference will depend on the required range; but in any event, the largest linearized range observed here (28 mV), from a combination of I _h and I _A , is significantly more costly than the 20 mV range that the I _h , I _NaP pair produces.

Appendices

Appendix A

The relationships of Table 5 are made more transparent by a few equations. Using the NaP-enhanced system as an example, the basic current equations (d = dendrite, s = synapses, and NaP = persistent sodium) and their derivatives are

$$\begin{array}{*{20}l} {{\begin{array}{*{20}l} {{I_{d} = g_{d} {\left( {V_{m} {\left( {g_{s} } \right)} - V_{d} } \right)},} \hfill} & {{\frac{{dI_{d} }}{{dg_{s} }} = g_{d} \frac{{dV_{m} }}{{dg_{s} }},} \hfill} \\ {{I_{s} = g_{s} {\left( {V_{m} {\left( {g_{s} } \right)} - V_{s} } \right)},} \hfill} & {{\frac{{dI_{s} }}{{dg_{s} }} = V_{m} {\left( {g_{s} } \right)} - V_{s} + g_{d} \frac{{dV_{m} }}{{dg_{s} }},} \hfill} \\ \end{array} } \hfill} \\ {{I_{{NaP}} = \overline{g} _{{NaP}} {\left( {V_{m} } \right)}{\left( {V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} } \right)} = \overline{g} _{{NaP}} \;f{\left( {V_{m} } \right)}{\left( {V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} } \right)}} \hfill} \\ {{\frac{{dI_{{NaP}} }}{{dg_{s} }} = \overline{g} _{{NaP}} {\left( {V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} } \right)}\frac{{df{\left( {V_{m} } \right)}}}{{dV_{m} }}\frac{{dV_{m} }}{{dg_{s} }} + \overline{g} _{{NaP}} \;f{\left( {V_{m} } \right)}\frac{{dV_{m} }}{{dg_{s} }}.} \hfill} \\ \end{array} $$

Remembering that, in steady-state, the sum of the currents is zero (i.e., I _s  + I _NaP  + I _d  = 0), the derivatives can also be combined and $\frac{{df\left( {V_{\text{m}} } \right)}}{{dV_{\text{m}} }}$ solved for such that

$$\begin{aligned} & \frac{{df{\left( {V_{m} } \right)}}}{{dV_{m} }} \\ & = - {\left[ {\frac{{g_{d} \frac{{dV_{m} }}{{dg_{s} }} + V_{m} {\left( {g_{s} } \right)} - V_{s} + g_{s} \frac{{dV_{m} }}{{dg_{s} }} + \overline{g} _{{NaP}} \;f{\left( {V_{m} } \right)}\frac{{dV_{m} }}{{dg_{s} }}}}{{\overline{g} _{{NaP}} {\left( {V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} } \right)}\frac{{dV_{m} }}{{dg_{s} }}}}} \right]} \\ \end{aligned} $$

(9)

Let V _s  = 0 and, in the linear range, let $\frac{{dV_{m} {\left( {g_{s} } \right)}}}{{dg_{s} }}$  = constant = K. Rearranging Eq. (9) gives

$$\frac{{df{\left( {V_{m} } \right)}}}{{dV_{m} }}\overline{g} _{{NaP}} {\left( {V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} } \right)}K = - {\left[ {g_{d} K + V_{m} {\left( {g_{s} } \right)} + g_{s} K + \overline{g} _{{NaP}} \;f{\left( {V_{m} } \right)}K} \right]}.$$

Taking the derivative with respect to V _m of both sides of this equation sets up the ordinary differential equation

$$\frac{{d^{2} f{\left( {V_{m} } \right)}}}{{dV^{2}_{m} }}{\left( {V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} } \right)} + 2\frac{{df{\left( {V_{m} } \right)}}}{{dV_{m} }} + \frac{2}{{\overline{g} _{{NaP}} K}} = 0,$$

which can be solved for f(V _m ). That is,

$$f{\left( {V_{m} } \right)} = - \frac{1}{{\overline{g} _{{Nap}} K}}{\left[ {V_{m} {\left( {g_{s} } \right)} + \frac{{C_{1} \overline{g} _{{NaP}} K + {\left( {V^{{Nernst}}_{{Na}} } \right)}^{2} }}{{V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} }}} \right]} + C_{2} ,$$

(10)

where C ₁ and C ₂ are constants. Finally, Eq. (10) and its derivative df(V _m )/dV _m can be substituted back into Eq. (9) which can then be solved for V _m in the linear range.

$$V_{m} = V^{{{\text{Nernst}}}}_{{{\text{Na}}}} + {\left( {g_{d} + g_{s} + C_{2} \overline{g} _{{NaP}} } \right)}K\;{\text{or}},\;{\text{equivalently}},$$

$$K = {\left| {\frac{{dV_{m} }}{{dg_{s} }}} \right|}_{{{\text{cst}}}} = \frac{{V_{m} {\left( {g_{s} } \right)} - V^{{{\text{Nernst}}}}_{{{\text{Na}}}} }}{{g_{d} + g_{s} + C_{2} \overline{g} _{{NaP}} }},$$

where C ₂ is a constant whose value is independent of g _s and V _m . The last equation shows that, in the linear range, since V _m changes linearly with g _s , the numerator is a linear function of g _s . This implies that the denominator also changes linearly (i.e., if g _s is doubled so must g _d and $\overline{g} _{{NaP}}$) to maintain dV _m (g _s )/dg _s = constant. The inverse relationship between dV _m (g _s )/dg _s and g _d or g _s or $\overline{g} _{{NaP}}$ is also apparent in the last equation. Equivalent equations can be derived for a system which includes I _A .

Appendix B

Table 6