Spiking neurons

Under resting conditions, a typical neuron is polarised at a membrane potential of around−70 mV. If its membrane potential rises above a threshold level (about −55 to−50 mV), it fires an action potential [43]. The dynamics of an action potential can be elaborated by non-linear conductance-based models, e.g. the Hodgkin-Huxley model [42]. Alternatively, it is also reasonable to model the sub-threshold dynamics and the firing behaviours separately, as the change in voltage is so rapid during a short time. Since this thesis mainly finds Green’s functions by assuming linear(ised) neuronal membrane dynamics, the second approach (to model the firing behaviours separately) can be easily incorporated into the framework. To be specific, Integrate- and-Fire (IF) models can be employed. Such models describe the two states of a neuron, firing and resting, independently by specifying the threshold voltage Vth. When the membrane potential eventually “integrates” toVth, it “fires” a spike and resets its value toVre. Although IF models are mathematical idealisations and thus lack biological details, they are useful because they are analytically solvable, even in cases of stochastic inputs, and therefore they have been widely used in analysis of emergent properties of neuronal circuits [43, 102].

To start with, consider a point neuron. The sub-threshold behavior of a leaky IF model is determined simply by the passive membrane (1.1) and the leakage current (1.3), that is, τm dV dt =El−V + I0 gl , (5.3) where τm = Cm gl . (5.4)

In addition, onceV ≥Vth, a spike is assumed to be generated andV is immediately reset toVre. It is worth noting the changes in V during a spike is not modelled, as the duration of a spike is considered extremely short. For any constant input current I0, Eq. (5.3) permits a fixed pointV =E0 whereE0 =El+I0/gl. If E0< Vth, the neuron is depolarised to a new equilibriumE0. IfE0 ≥Vth, its membrane potential keeps increasing towardsE0 but always reaches the threshold and then resets first, that is, the neuron fires periodically. Without loss of generality, chooseV(0) =Vre

and the solution to Eq. (5.3) can be found as

V(t) =E0+ (Vre−E0)e−t/τm_{. (5.5)}

V(T) = Vth then gives the duration T for the potential to reach the threshold, explicitly, T =τmln E0−Vre E0−Vth . (5.6)

SinceT is the period of firing, the firing rate can be easily found asT−1.

To make the model more realistic, an additional function for the spiking regime can be considered instead of the instant reset, while the sub-threshold regime de- scribed by Eq. (5.3) is unchanged. Explicitly, the function can be defined as

V(t) =hs(t−tis), (5.7) fort∈(ti_s+Ts] and anyi∈ {1,2,3, . . .}, wheretis is the time when thei-th spike is initiated andTsis the duration of a spike. In addition,hs(0) =Vthandhs(Ts) =Vre. Thus, when V < Vth the membrane potential in this extended model is described by Eq. (5.3), but once it reaches the threshold it follows Eq. (5.7) until the spiking regime ends (afterTs). It is straightforward to see that the original leaky IF model with the instant reset can be recovered by the new model with the limit Ts → 0. The new model has a period ofT+Ts, which gives a firing rate of (T+Ts)−1. The definition ofhsbecomes really important and useful when neurons with morphologies are considered. For example, Schwemmer and Lewis [103] considered a “ball-and- stick” model, and only the soma is excitable, that is, if the somatic membrane potential reaches the threshold, it starts to vary as Eq. (5.7) specifies. If the voltage is reset instantly, i.e. Ts → 0, the somatic membrane potential is discontinuous in t at the spiking times ti_s. However, there is always a boundary condition at the location where the soma and the dendrites are attached together that requires the somatic and the dendritic membrane potentials to be equal (i.e. the continuity of membrane potentials, see §1.2.4). A contradiction arises because the dendritic potential is continuous in t (even at ti_s). Since Green’s functions are found by the sum-over-trips framework, it is simple to compute responses given any inputs. It is then trivial to check for the soma whether or not their membrane potentials are above the threshold. If so, it fires a spike, which can be treated as injecting a somatic input current. The entire voltage response profiles after the occurrance of spiking events are thus recursively updated. Such a simulation procedure saves computational cost, because given the fixed time window of voltage profile it only

needs to be updated as many as the number of spiking events, comparing to any numerical simulation that updates every time steps.

Another example is to model dendritic spines as IF active points. Dendritic spines are extensions on dendrites that connect post-synaptic neurons to axons of pre-synaptic neurons. Many chemical synapses can be found on the head of them, and thus they are closely related to spikes. Spine heads are excitable and can thus be modelled by IF active points [79] (as the soma in the last example), whereas Hodgkin-Huxley models are also applicable [104]. Spine necks can be simply mod- elled as passive resistors. Assuming spines distributed in a discrete and uniform density along a dendritic branch, we obtain the Spike-Diffuse-Spike (SDS) models [79–82]. Comparing to brutal-force simulations, solving for Green’s functions be- tween two such spines first allows one to repetitively add the responses directly into the entire system whenever the voltage of any spine reaches the IF threshold.

In addition, since the Fire-Diffuse-Fire (FDF) models for intra-cellular calcium releases and waves [83, 84] are similar to the SDS models in their mathematical expressions, the approaches taken in this thesis can be also employed for the FDF models [85]. IF active points can also be added onto the model of the nerve ring discussed in§2.4.3. Such a theoretical model can assist experimentalists understand- ing nerve rings in jellyfishesPolyorchisandAglantha, their central nervous systems, because these creatures are small in size and it is difficult to obtain intra-cellular recordings from their neurons [77, 78].