A commonly accepted truth that ”transient electrical signals are particularly important for carrying time-sensitive information rap- idly and over long distances. These transient electrical signals—receptor potentials, synaptic potentials, and action potentials—are all produced by temporary changes in the electric current into and out of the cell, changes that drive the electrical potential across the cell membrane away from its resting value.”[41]” ”Most voltage-gated channels, in contrast, are closed when the membrane is at rest and require membrane depolarization to open.”
By putting together the operating stages (as we have mentioned: in a well-defined order), one receives the characteristic process of neuronal operation (an operating cycle): the stage variable changes in a well-defined pace in the function of the local time. As depicted in Fig. 1.6, the stage variable can be well observed inside the cell, furthermore, it has a well-measurable effect outside the cell. The notion of neuronal operation is the current pulse (called ’Action Potential’) has a central role in neural operation. Here we discuss concepts of its production, while its sending and receiving in the next sections. The stages we have discussed previously are are color-coded in the diagram line. The stage variable in the ”Computing” stage is observable only inside the cell. After the beginning of stage ”Delivering”, the characteristics of the emitted charge pulse are well observable also outside the cell. The figures on the axes are approximately correct: the measurable voltage change is up to several dozens of millivolts, and the time scale is up to several milliseconds.
As we discussed in section 1.7.1, at the beginning of an operating cycle, synapses are open and some input pulses (gradient steps) increase the membrane’s potential [80, 81] (the green section of the broken line). After exceeding the membrane’s threshold voltage (the dotted orange line), the synapses’ gating mechanism closes the current inputs and the membrane’s rush-in mechanism begins to work due to opening voltage-controlled ion channels in the membrane’s wall. The effect extends over the surface in an avalanche-like way [82]. The voltage increases until all ion channels get opened. The ion channels close after a very short period and the neuronal circuit continues its operation with discharging the condenser (the red section of the broken line). Given that the condenser stores part of the received charge, the capacitive current decreases and later reverses and reverts also the resulting current (and, consequently, the measurable voltage; this effect is observed as the membrane gets hyperpolarized). In this period (the blue section of the broken line) the synapses are open again and the received synaptic input gradients contribute again to the membrane’s voltage. That is, a new cycle (this time starting with a potential differring from the resting value) can start and that residual potential acts as a memory (the details of the electrical processes are discussed in section 2.5.6). Notice that all mentioned events are spatiotemporal; most noticably, the arrival of the synaptic inputs to the synaptic terminal following the absolute refractory period is much earlier than they are observable as an increase in the potential value (measured at the AIS). Also notice that there is no direct connection between the input and output voltages: the circuit fundamentally changes the neuronal output.
With reference to Fig. 1.6, we subdivide neuron’s operation to three stages (green, red, and blue sections of the broken diagram line), in line with the state machine in Fig. 1.4. We start in stage ’Relaxing’ (it is a steady-state, with the membrane’s voltage at its resting value). Everything is balanced, the synaptic inputs are enabled. No currents flow (neither input nor output; at this point we do not consider the dynamic balancing current discussed in section 2.7), since all component have the same potential, there is no significant driving force for an output current.
As the figure suggests, an external invasion, typically, an electric voltage on the intracellular side, changes the balanced state and due to the parameters are linked, changes all concentrations. When moving the system out of its balanced state, in any way, a driving force appears that moves the system towards finding a new balanced state. However, recall that the ions are slow, so the changes are not instant.
The neuron has a stage variable (the membrane potential) and a regulatory threshold value. There exists a threshold for voltage gradient instead of the membrane’s voltage itself (the voltage gradient provides a ’driving force’). As we detail in section 2.8.5, the voltage sensing is based on voltage gradient sensing, which phenomenon correlates with the value of membrane’s voltage. Given that physiological measurements (such as clamping) suppress the gradient, and only the voltage is measured in a ’freezed’ state, this difference has remained hidden. Crossing the membrane’s voltage threshold value upward and downward causes a stage transition from ”Computing” to ”Delivering” and from ”Delivering” to ”Relaxing”, respectively. Another role of that regulatory value is to open/close the input synapses. Furthermore, when the value exceeds the threshold, an intense current starts to charge up the condenser, that later discharges. We show that, although the change correlates with the value of membrane’s voltage, the neuron’s membrane actually senses the voltage gradient.
Given that the neuron’s operation resembles an oscillator, the capacitive current of the condenser changes its direction, leading to changing the potential relative to the charge-up potential value to a value of opposite sign. The time constant of the oscillator is set so that the rushed-in current generates a nearly critically damped oscillation (with a damping parameter about ).
Notice that all these processes happen with well-defined speeds, i.e., the different stages have well-defined temporal lengths. The length of period ”Delivering” is fixed (defined by physiological parameters), the length of ”Computing” depends on the activity of the upstream neurons (furthermore, on the gating due to the membrane’s voltage). Due to the finite speed, we discuss all operations in neuron’s own ”local time”.
When the membrane’s voltage decreases below the threshold value, the axonal inputs are re-opened, that may mean an instant passing to stage ”Computing” again. The current stops only when the charge on the membrane disappears (the driving force terminates), so the current may change continuously, changing the voltage on the circuit’s output. The time of the end of operation is ill-defined, and so is the value of the membrane’s voltage at the time when the next axonal input arrives. The residual potential acts as a (time-dependent) memory, with about a lifetime; see Fig. 1.6.
With putting together the operating stages (as we have mentioned: in a well-defined order), one receives the characteristic process of neuronal operation (an operating cycle): the stage variable changes in a well-defined pace in the function of the local time. As depicted in Fig. 1.6, the stage variable can be well observed inside the cell, furthermore, it has a well-measurable effect outside the cell. The notion of neuronal operation is the current pulse (called ’Action Potential’) has a central role in neural operation. Here we discuss concepts of its production, while its sending and receiving in the next sections. The stages we have discussed previously are are color-coded in the diagram line. The stage variable in the ”Computing” stage is observable only inside the cell. After the beginning of stage ”Delivering”, the characteristics of the emitted charge pulse are well observable also outside the cell. The figures on the axes are approximately correct: the measurable voltage change is up to several dozens of millivolts, and the time scale is up to several milliseconds.
As we discussed in section 1.7.1, at the beginning of an operating cycle, synapses are open and some input pulses (gradient steps) increase the membrane’s potential [80, 81] (the green section of the broken line). After exceeding the membrane’s threshold voltage (the dotted orange line), the synapses’ gating mechanism closes the current inputs and the membrane’s rush-in mechanism begins to work due to opening voltage-controlled ion channels in the membrane’s wall. The effect extends over the surface in an avalanche-like way [82]. The voltage increases until all ion channels get opened. The ion channels close after a very short period and the neuronal circuit continues its operation with discharging the condenser (the red section of the broken line). Given that the condenser stores part of the received charge, the capacitive current decreases and later reverses and reverts also the resulting current (and, consequently, the measurable voltage; this effect is observed as the membrane gets hyperpolarized). In this period (the blue section of the broken line) the synapses are open again and the received synaptic input gradients contribute again to the membrane’s voltage. That is, a new cycle (this time starting with a potential differring from the resting value) can start and that residual potential acts as a memory (the details of the electrical processes are discussed in section 2.5.6). Notice that all mentioned events are spatiotemporal; most noticably, the arrival of the synaptic inputs to the synaptic terminal following the absolute refractory period is much earlier than they are observable as an increase in the potential value (measured at the AIS). Also notice that there is no direct connection between the input and output voltages: the circuit fundamentally changes the neuronal output.
Sending AP begins when the membrane’s potential exceeds the threshold and terminates when it drops below the threshold. The time of the stage ”Delivering” is entirely determined by the parameters and of the neuronal oscillator, so all outgoing spikes have the same shape. However, depending of the time of the previous spike (or, more precisely, the value of the membrane’s potential at the moment of the start of ”Delivering”), the shape may be apparently different. The measurable potential is the sum of the ”tail” of the previous spike plus the front of ”this” spike; and both of them have their temporal course (the currents that evoke those voltages are slow). The effect of gating can be observed with a time delay at the AIS (the slow current entering through the synaptic terminals needs time to reach the AIS), and the value of that delay depends on the geometry of the neuron, mainly on the position of the synaptic terminal. On its ”local time ” scale, the AP starts when the first exciting synaptic pulse arrives, usually (due to exceeding the threshold for the voltage gradient) leading to starting the neuron’s rush-in current.
The native input arrives through the synaptic terminals, at the time determined by the upstream neuron (in the sense that at what absolute time it sends the spike and how long the spike travels). The time window of the ”Computing” period opens when the first input arrives and so the further inputs arrive at a later time. The time window ends when the membrane’s voltage exceeds its threshold. Given that the computing time is in the order of and the total length of an AP is in the order of up to , only the first temporal part of the received spike can be processed and can contribute to the result: as the membrane’s potential increases, the neuron closes its synapses. That means that the neurons coooperate with their upstream neurons: the contributions to their membrane’s charge through their synapses change even between the adjacent spikes. Even, the composition of the sum may change, depending on in which order the input spikes arrive.
During regular neuronal operation, spikes arrive through synapses, and their effect can also be measured as a PSP. When a spike (evoked by a single AP, elicited by current injection in presynaptic cells) [83] arrives at a synapse, it can be represented that a (short pulse of) “slow” current arrives through the axon. However, the inflow of the axonal current is “slow”, and a “critical mass” of ions is needed to start a well-defined current inflow into the membrane, so neuronal arborization [84] takes place, forming an “ion buffer”. If a current arrives through the axon, when entering the arbor, the cross section suddenly increases, so suddenly decreases; see Eq.(2.25). The arbor buffers the charge received in the spike. The mechanism that the ion current ’takes away’ the ions does not work in the arbor. The ions can move under the voltage gradient resulting from the mutual repulsion and the current drain towards the membrane. The drain current (into the membrane) is proportional to the voltage gradient between the arbor and the membrane, giving a natural explanation how the membrane’s potential controls synaptic contributions, furthermore why the potential increase in the ”relative refractory” period changes with the membrane’s potential. For details, see section 1.3.5 and Figure 3.22. Whether the buffer is filled or empty explains that ‘both sodium and potassium conductances increase with a delay when the axon is depolarized but fall with no appreciable inflexion when it is repolarized’ [9].
The arbor essentially (and anatomically) belongs to the axon, but its functionality is also similar to that of the membrane. It plays a vital role in the information processing in the brain [85]: defines the crucial input parameter “time of arrival of a spike”, makes the intensity of synaptic inputs nearly independent from the shape of the spike (less depending on the presynaptic neuron; important for the cooperation); furthermore, links adjacent spikes, providing a neuronal memory. The buffering effect may be seen as “making a hole in the membrane” [81]: exceeding a critical mass (charge in the arbor) may start an intensive current into the membrane and manifest in a sudden change, see section 3.7.6. The shape of PSP also plays a vital role in synchronizing neurassemblies [15, 63].
The buffering changes the shape of the received AP: it integrates the input axonal current and distorts the received AP’s shape toward a PSP: there is no AIS in the axon (no oscillator). Most schematic figures showing signal transmission from a presynaptic neuron to a postsynaptic neuron miss the point that at the synapse, the AP appears as having a different shape. Furthermore, they misidentify the temporal length of AP essentially to a period between the beginning of the charge-up to the end of reaching the hyperpolarization peak voltage (comprising the ’Delivering’ stage and some part of ’Relaxing’). Our discussion shows that the stage ’Computing’ (reaching the threshold potential) and ’Relaxing’ (the long tail after hyperpolarization) are a vital part of AP. The former is the result of the neuronal computation and the latter (among others) provides short-term neuronal memory for neuronal cooperation.
The intense current from this buffer starts to charge the membrane and discharge the arbor. We arrive back at the case we saw in the case of a clamping where a ”slow” axonal input current from the arbor arrived at the membrane at its junction. It flows with its finite speed on the membrane’s surface, while, at the same time, the newly created potential decays exponentially. Initially, while the buffer is charging, the current increases exponentially as the spike arrives, manifesting in the observable PSP. We can validate our model-based hypothesis by fitting experimental data; see section 3.9.