OLOCAU DEL REY
IGLESIA PARROQUIAL DE SANTA MARÍA DEL PÓPULO
To build a model neuron, we can consider a current clamp experiment such as the one shown in Figure 1.73. In this section, we will study the neuron during quiescence, so let us consider only the values of input current I that are not suffi- ciently depolarizing to initiate action potentials (i.e. the current is below rheobase, Iρ). For a particular current I, the membrane potential of the neuron V evolves towards a fixed value and then does not move away from this value. From a dy- namical systems perspective, this fixed value would be described as an invariant set, formally defined as
Definition 2. A set M is invariant if for all x0 ∈ M , ϕ(t, x0) ∈ M for all t -
Glendinning(1994).
That is, a flow initially in an invariant set M remains in M for all time (Strogatz,
2014). In the case of the neuron at rest, the membrane potential reaches a fixed value and then does not chage, meaning this invariant set consists of a single
3The data presented inFigure 1.7was collected by the author of this thesis from a neuron in
0 Time [s] 1 -60 -40 -20 0 20 40 V [mV] Data 0 Time [s] 1 -20 0 20 40 I [pA] 0 Time [s] 1 -60 -40 -20 0 20 40 V [mV] Simulation
*
Figure 1.7:Bifurcations in neural dynamics - from resting to spiking. This fig- ure shows an example of a current clamp experiment in a neuron. A step current (bottom left) is applied across the membrane of the neuron, and the membrane potential V of the neuron is recorded (top left). For currents I below some critical value Iρ, called rheobase, V reaches a steady state. Above this critical value, the neuron fires action potentials. Therefore a key feature of mathematical models of neurons is that they may be described by ˙V = f (V ; I) and undergo a bifurcation at I = Iρ. A simple example of this is the quadratic integrate-and-fire (QIF) neuron (seesection 1.4.2 for a description of the QIF neuron and the integrate-and-fire reset condition), in which f (V ; I) is a quadratic function. The plots at the top right show that in such a system, there exists a stable and an unstable steady state for I < Iρ, which collide to form a non-hyperbolic equilibrium at I = Iρ, and are abolished for I > Iρ. Therefore this model of a neuron transitions to spiking via a saddle node bifurcation (described further in section 1.4.1). By resetting V to some value Vreset when it reaches a value Vpeak, regular spiking can be achieved for I > Iρ. An example simulation of the data using a quadratic integrate-and-fire neuron is shown in the bottom right figure in black, overlaid on the data (grey) for comparison purposes.
Dynamical systems and qualitative modelling of the neuron
point x∗ ∈ Rm. This particular type of invariant set is known as a steady state, fixed point, or equilibrium of the system, which satisfies
f (x∗) = 0. (1.2)
Therefore a neuron model ˙x = f (x; I) with parameter I < Iρ might be chosen such that there exists some x∗ that satisfies f (x∗; I) = 0to qualitatively describe the non-spiking dynamics of the membrane potential.
However, this condition alone is not sufficient to ensure that the behaviour of the steady state matches the behaviour of the quiescent neuron. Neurons reg- ularly receive small perturbations away from the steady state due to stochastic ion channel gating and post-synaptic potentials (White et al., 2000). An exam- ple of the latter is marked by a red star in Figure 1.7. Note that following the perturbation, the membrane potential stays in the neighbourhood of the steady state and eventually returns to it. From a dynamical systems perspective, this means the steady state is stable. Whilst there exist various definitions of stability (Glendinning,1994), the most useful definition in the context of this thesis is Definition 3. An invariant set M is stable if it satisfies both of the following con- ditions:
i. Lyapunov stability: Flows in the neighbourhood of M remain in the neighbour- hood of M ,
ii. Asymptotic stability: There exists some neighbourhood U ⊃ M where all flows initially in U tend to M as t → ∞.
-Kuznetsov(1998);Strogatz(2014).
Stability of a steady state can be established by the following theorem:
Theorem 1. Let x∗ be a steady state of a dynamical system of the form Equa- tion 1.1. Let A ≡ J (f )|x∗ be the Jacobian matrix of f (x) evaluated at x∗. Then
x∗ is stable if all eigenvalues λj ∈ C, j = 1, 2, . . . , m of A satisfy <(λj) < 0. -
Kuznetsov(1998)
This outcome is a result of the Hartman-Grobman theorem, which states that Theorem 2. Hartman-Grobman Theorem. Let x∗ be a hyperbolic (all eigen- values of J (f )|x∗ have non-zero real part) steady state of a non-linear dynamical
system of the formEquation 1.1. Then the non-linear system is locally topologi- cally equivalent near x∗ (seeDefinition 4) to the linearized system ˙ξ = J (f )|
x∗ξ. - Kuznetsov(1998)
A more rigorous statement and proof of this theorem can be found in Robinson
Definition 4. A dynamical system is topologically equivalent to another dynam- ical system if there exists a homeomorphism h : Rm → Rm that maps orbits of the first system onto the second system. - (Kuznetsov,1998)
Local topological equivalence near a steady state restricts this to a small neigh- bourhood of the steady state.
Perturbations to a linear system of differential equations ˙ξ = Aξ grow or decay proportional to exp λjtin the direction of the jth eigenvector of A (Strogatz,2014), where λj is the jth eigenvalue of A, meaning that the linearization of a dynamical system about a steady state satisfies Definition 3 if <(λj) < 0 ∀j. Therefore the steady steady state of the original non-linear system is also stable, by the Hartman-Grobman theorem.
We have now described properties of a simple dynamical system describing a neuron in the non-firing regime when a small current I < Iρ is applied. When the input to the neuron is sufficiently depolarising (I > Iρ;Figure 1.7), the neuron enters a spiking regime in which action potentials are fired and the membrane potential no longer tends to a fixed value. In dynamical systems theory terms, this means that at I = Iρ the steady state is either abolished or loses stability. This sudden qualitative change in dynamics of the system as a parameter is varied is known as a bifurcation (Strogatz,2014), more formally defined as
Definition 5. A bifurcation is the appearance of a topologically nonequivalent phase portrait under variation of parameters. -Kuznetsov(1998)
When describing bifurcations, it is often useful to discuss the normal form of the bifurcation,
Definition 6. The normal form of a bifurcation is locally topologically equivalent to any generic system which undergoes the same bifurcation in the neighbour- hood of the bifurcation parameter -Kuznetsov(1998);Izhikevich(2007);Strogatz
(2014).
Figure 1.7 shows an example of a bifurcation in a one-dimensional system which satisfies the case in which the steady state is abolished as I is increased above Iρ, replicating the loss of stable steady state in the membrane potential of the neuron. This bifurcation, called the saddle-node (SN) bifurcation, has the normal form
˙x = x2+ µ (1.3)
(Strogatz, 2014). For µ < 0, there is a stable steady state at x∗− = − √
−µ and an unstable steady state at x∗+ = +√−µ. As µ → 0 from the negative side, these steady states move together and at µ = 0 collide to form a single non-hyperbolic steady state. For µ > 0, there are no real solutions so no steady states.
Dynamical systems and qualitative modelling of the neuron
The saddle node bifurcation qualitatively captures the loss of quiescence in the neuronal membrane potential. To obtain a more quantitatively accurate rep- resentation of the membrane potential, a transformation of variables can be made such that the steady states match the values of the membrane potential and the bifurcation occurs at µ = Iρ, whilst remaining topologically equivalent to Equa-
tion 1.3(Izhikevich,2007;Gerstner et al.,2014).