3.7.4 Neural currents

When patching, a current is directly introduced to the neuron’s body. In the case of a constant current where $I=\frac{dQ}{dt}$, the voltage increase $dV$ on the capacity $C$ of the membrane is $\frac{dQ}{C}=\frac{I\ast dt}{C}$, so

The direct constant current input $\frac{d}{dt}{V}_{PATCH}$ to the neuron cell body is a simple constant current that causes a constant membrane’s voltage derivative contribution. However, the currents are not necessarily constant. If the artificial current follows a math function, the time derivative of that function should be used. In the case of a native current (i.e., receiving a spike form a presynaptic neuron), the received input has the form of PSP, where the time derivate can be well approximated by a steep exponential function. One must be careful that (step-like) sudden changes may produce very steep spikes (see the wave forms in Table 2.1 on differentiating a square wave function); furthermore, as we discuss in section 3.9.3, the step-like concentration change causes exactly the same change in the output voltage, only the time scale differs in a factor of ${10}^6$.

When clamping, the current is injected through an axon, by switching a clamping voltage to the axon. Given that the current is delivered through the axon, the mechanisms described in section 3.9.2 must be considered. The current at the switch ON/OFF events behaves as a step function; that is, it produces a saturating and a discharging current, respectively. The switch-on effect is known also in technical electricity; in biology its time constant is in the order of $1ms$, that is drastically influences the measured biological processes, see Figure 3.19. Recall that in the case of clamping, the derivative contains an exponential function. In the case of patching, the derivative is the derivative of a (nearly) square-wave function. For a discussion of the measured result, see section 3.8.5.

The AIS represents a non-distributed resistance $R_M$, and the current flowing through it is

(it is an outward current, so it is negative), and its voltage time derivative is

Notice that this current depends on $C_M{R}_M$, all others on $C_M$. (This current was mis-identified by HH as ’leaking current’: if no other current/voltage derivative is present, the membrane discharges. In resting state the derivative is zero: the condenser is charged up and no leaking current flows.

In the case of those currents, as we discussed in the cases of membrane and axon, a saturation-type function multiplied by a decay-type function describes the current, so the voltage derivative is

The same voltage derivate (with different parameters $a$ and $b$) is valid for $\frac{d}{dt}{V}_M(t)$ due to the membrane rush-in current (as discussed above, the voltage derivate is proportional to the current through a factor $1/C_M$). See also Figure 3.15.

In the native case (the membrane’s voltage created instantly and then no external invasion happens), the resulting voltage derivative is

Figure 3.11 shows the functional forms of $V_M(t)$ and $\frac{d}{dt}{V}_M(t)$ ( PSP current and its voltage derivative) at some reasonable parameter values $a$ and $b$). (Notice that the front of an arriving spike, as well as at the beginning of clamping, the front is almost clearly exponential.) Notice the sudden change of the derivative after the output (spike delivery) begins: the exponential increase of $V_M(t)$ really causes a steep change in its derivative at low time values. For different values of parameters $a$ and $b$, a variety of function shapes describing APs can be created, see Figure 3.10 and also Figure 3.15.

In the most complex case, the time derivative of voltage we need to work with is

The first term is always present. The second term only if previously exceeding the threshold value caused by membrane’s charge-up (an instant effect). The third term changes during the stages of operation, as we describe below. The last term is an ”artificial” contribution (and so: it depends on experimental settings), but it is frequently used in experimental research. Notice that whether voltage or current clamping is applied, it only means what the experimenter keeps constant; it acts with its voltage derivative. The same holds for the mathematical form of the used current/voltage.

The equation enables us to understand the experience that the shape of the AP is always the same. More precisely, the integrals of the contributing $\frac{d}{dt}V(t)$ terms remain the same. Furthermore, if the contributors remain the same, the resulting shape also remains the same. Of course, only in steady state. It changes, if the next spike arrives before the resting potential restored, or synaptic input arrives when synaptic inputs are enabled, or the artificial current changes.

The neuron’s electric operation comprises several stages, and the different physical phenomena produce different currents in those stages. The stages of neuronal operation, and the presence of slow and fast currents, furthermore the gating mechanisms significantly shade the picture.

As we introduced, the ion currents are ’slow’ if they arrive through the axon (as [9] measured, an apparent ’delay’ can be observed between the voltage and the current).

The ’artificial’ contributions $\frac{d}{dt}{V}_{CLAMP}$ and $\frac{d}{dt}{V}_{PATCH}$, of course, depend only on the investigators and no additional (stage-dependent) rule is followed (although the delay may apply).

The contribution $\frac{d}{dt}{V}_{AIS}(t)$ is always on; the neuron all the time, independently from its history, operating stage and its inputs, attempts to restore its resting potential. The $I_{AIS}$ is active all the time, active all the time. However, it is not a ”leaking current”. It is proportional to the difference of the membrane’s potential above the resting potential. In resting state, its value is zero, see The mechanism in resting state is different.

The contribution $\frac{d}{dt}{V}_M(t)$, once ’DeliveringBegin’ issued, will not be stopped (except ’Synchronize’) until the membrane voltage drops below the threshold value. If the artificial currents are too high (see 3.15), the stage ’Delivering’ may last forever.

The contributions $\frac{d}{dt}{V}_{SYN,i}(t)$ are only enabled when the membrane’s voltage is below the threshold level. The amplitude of the current/voltage derivative depends on the membrane’s voltage. The synaptic inputs $I_{SYN,i}$ are active only in the charge-up and ’relative refractory’ period. Actually, when the membrane potential is kept above the threshold value: the ions cannot enter the intracellular space against the higher membrane potential: the ’normal’ inputs can be blocked [147]. See also Figure 1.6.