ion concentrations during neural activity

Below we will perform a calculation to assess the extent of sodium ion concentration change during an action potential.

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We can model biological membranes as a simple equivalent circuit with a capacitor and resistor in parallel:

• the capacitor represents the lipid bilayer, responsible for storing charge across the plasma membrane,
• the resistor represents the combined ion permeation pathways (e.g. ion channels and transporters) in the membrane.

Change in the entire neuron

Membrane charge transfer

The charge separated across the biological membrane is given by

$$Q=CV=\left(10^{-6}\frac{\text{F}}{{\text{cm}}^2}\right)\left(0.1V\right)=10^{-7}\frac{\text{C}}{{\text{cm}}^2},$$

where:

• $C$ is the specific membrane capacitance
• $V$ is the change in membrane potential which occurs during an action potential

Since each ${\text{Na}}^{+}$ ion carries an elementary charge of $1.6\times 10^{-19}\text{C}$, the number of ions per $\mu {\text{m}}^2$ is:

$$\frac{1\times 10^{-15}\text{C}/\mu {\text{m}}^2}{1.6\times 10^{-19}\text{C/ion}}\approx 6250\text{ions}/\mu {\text{m}}^2.$$

Total ion entry over the neuron

Assuming a roughly spherical neuron with a diameter of $10\mu \text{m}$, the surface area is:

$$A\approx 4\pi r^2\approx 4\pi (5\mu \text{m}{)}^2\approx 314\mu {\text{m}}^2.$$

Thus, the total number of ${\text{Na}}^{+}$ ions entering during the spike is:

$${\text{Total Na}}^{+}\approx 314\mu {\text{m}}^2\times 6250\text{ions}/\mu {\text{m}}^2\approx 2\times 10^6\text{ions}.$$

Relative change in concentration

The cytoplasmic volume of the neuron is approximately:

$$V\approx \frac{4}{3}\pi r^3\approx \frac{4}{3}\pi (5\mu \text{m}{)}^3\approx 524\mu {\text{m}}^3.$$

With a resting intracellular ${\text{Na}}^{+}$ concentration of $10\text{mM}$, the total number of ${\text{Na}}^{+}$ ions is roughly:

$$\sim 3.2\times 10^9\text{ions}.$$

The fractional increase is then:

$$\frac{\Delta N}{N}\approx \frac{2\times 10^6}{3.2\times 10^9}\approx 0.06\%.$$

This small percentage supports the assumption that bulk intracellular ionic concentrations remain essentially constant during an action potential.

Change in a single spine

Resting sodium content

A typical pyramidal cell spine head with a volume of $V_{\text{spine}}\approx 0.02\mu {\text{m}}^3$ is estimated to contain roughly

$$\sim 1.2\times 10^5{\text{Na}}^{+}\text{ions}$$

at a resting concentration of $10\text{mM}$ (Eberhardt et al., 2022).

Ion entry during synaptic event

During an excitatory postsynaptic potential (EPSP), suppose a synaptic current of 23 pA flows for 12 ms (Cornejo et al., 2022; Acker et al., 2016). The total charge delivered is:

$$Q=It=(23\times 10^{-12}\text{A})(12\times 10^{-3}\text{s})\approx 2.76\times 10^{-13}\text{C}.$$

Converting this charge into the number of ${\text{Na}}^{+}$ ions:

$$n_{\text{ions}}=\frac{2.76\times 10^{-13}\text{C}}{1.6\times 10^{-19}\text{C/ion}}\approx 1.73\times 10^6\text{ions}.$$

Relative change in the spine

The ratio of the ions entering during the synaptic event to the resting content is:

$$\frac{1.73\times 10^6}{1.2\times 10^5}\approx \mathrm{14.4.}$$

This implies that the local sodium concentration in the spine could, in principle, increase from 10 mM to approximately:

$$10\text{mM}\times 14.4\approx 144\text{mM},$$

a dramatic increase compared to the whole neuron.