 Solution

Assume the structural surface charge density stays fixed while a divalent chloride is added, without changing $T$ or $ϵ$. Let $v=e{\varphi }_0/(k_{B}T)$ and $\delta c$ be the increment of the bulk divalent concentration. Electroneutral addition also increases bulk chloride by $2\delta c$. Varying the squared Grahame equation at fixed $\sigma$ givesAt a strongly negative wall, coions are small. Dropping their contribution to the denominator and $2e^v$ in the numerator gives the requested leading approximationThe $-3$ includes the change in bulk osmotic concentration of one added divalent counterion and its two monovalent coions. The numerator's increment is a bulk concentration; the denominator contains surface concentrations.^[ref] Extra counterions screen the wall and make its negative electric potential less negative. At fixed bulk calcium, $c_{\mathrm{Ca}}(0)=c_{\mathrm{Ca}}^{\infty }{e}^{-2{\varphi }_0/V_T}$ therefore decreases. Under the stipulated proportionality, synaptic activity decreases as well; this argument isolates electrostatic accumulation rather than assuming a separate specific magnesium effect on the ion channels.

For $\delta c=5\mathrm{mM}$, the printed linear approximation gives $\delta {\varphi }_0=9.53434\mathrm{mV}$. Linearizing the activity too gives $\delta A/A=-2\delta {\varphi }_0/V_T=-0.7422$, or approximately a $74\%$ reduction. Exponentiating this approximate voltage instead gives $A^{\prime }/A=0.4761$. Neither is a controlled finite-increment prediction: the added divalent concentration is 2.5 times its initial value.

A consistent finite-change calculation is possible within the same Poisson-Boltzmann equation. Define, using $\mathrm{mM}$ throughout,Fixed $\sigma$ requires $S_5(u^{\prime })=S_0(2.84129)$. The physical positive root is $u^{\prime }=2.58241$, givingThus the self-consistent finite-change estimate is about a 40.4% reduction. Both estimates rely on fixed surface charge, ideal ionic activities, and unchanged bulk calcium; the limitations of the large local concentrations remain.