Spike adding through flip bifurcations and the flip cascade

Often in models of bursting neurons there are sudden changes in the number of spikes per burst as a parameter is smoothly varied. An important mechanism by which this can occur is the flip bifurcation (Channell et al., 2007; Barrio and Shilnikov, 2011; Tsaneva-Atanasova et al., 2010). The flip bifurcation is a local bifurcation of periodic orbits, meaning it can be studied locally on fixed points of the Poincar ´e return map corresponding to periodic orbits of the flow. The normal form of the flip bifurcation is the map

x 7→ P (x) = −(1 + α)x + x3, (1.11)

for |α|  1 (Kuznetsov, 1998). There is a fixed point at ¯x = 0 which is stable for α < 0and unstable for α > 0, hence it is clear a bifurcation occurs at α = 0. To fully understand the dynamics of the flip bifurcation, we can also study the second iterate x 7→ P2_{(x) ≡ P (P (x)), which naturally has the same fixed point at the} origin with the same stability as P (x). However, for α > 0, P2_{(x)additionally has} fixed points at x ≈ ±√α(Kuznetsov,1998). The points are stable and correspond to a period two cycle of the map x 7→ P (x). For this reason, the flip bifurcation is often also called the period doubling or spike adding bifurcation. The bifurcation diagram for P2_{(x)is shown inFigure 1.11A.}

An interesting phenomenon is the flip cascade (Kuznetsov,1998), which is an additional route to changing the numbers of spikes in a burst of neurons through chaos (Channell et al., 2007; Tsaneva-Atanasova et al., 2010), and plays a key role in spike-adding in chapter 2. In a flip cascade, a series of flip bifurcations occur, changing the period of the orbit on the map until the dynamics become chaotic. An example of this is shown for the logistic map x 7→ p(x) = αx(1 − x) (Strogatz, 2014) in Figure 1.11B. Crucially, there are intermittent periods of regular, non-chaotic orbits on the map, with a period that differs from the period of the previous non-chaotic orbit (Strogatz,2014). Therefore a flip cascade through chaos can act as a path between orbits of the map with different periods.

Figure 1.11: Flip bifurcations. (A) Bifurcation diagram for the flip bifurcation. Solid/dashed black lines are stable/unstable fixed points on the second iterate of the map (P2_{(x)). Since the flip bifurcation holds in the limit of small |α|} (Kuznetsov, 1998), shown here is the case for |α| ≤ 0.2 and neglecting O(x5₎ terms (Kuznetsov, 1998). (B) Orbit diagram demonstrating a flip cascade into chaos and intermittent regions of non-chaotic orbits, demonstrating how a flip cascade can be a path through chaos to a change of period of an orbit. For demonstration purposes, the orbit diagram shown here is for the logistic map.

1.5

Biophysical modelling of the neuron

Section 1.4introduced qualitative descriptions of neuronal excitability from a dy- namical systems perspective, and introduced some low dimensional models of the neuron based upon this bifurcation behaviour. This type of model is known as a phenomenological model, meaning it is grounded in replicating dynamic phe- nomena without a detailed biophysical description. Such models are highly useful for simulating large populations of synaptically connected neurons in parallel to study network dynamics (Izhikevich, 2003;B ¨orgers and Kopell,2005). However, phenomenological models are of limited use for understanding ionic mechanisms of neuronal dynamics - for example how changes to the flow of charged ions across the membrane alter the dynamics of neurons in disease. For this purpose,

Biophysical modelling of the neuron A _𝐼 𝐶 𝑔_$%𝜓_$% 𝑔_'𝜓_' 𝑔₍ 𝐸$% 𝐸' 𝐸( B C D E F Extracellular Intracellular

Figure 1.12: Quantitative description of action potential generation in the Hodgkin-Huxley model. (A) The cell membrane acts as a resistor-capacitor cir- cuit, with the membrane acting as a capacitor and the voltage gated ion channels acting as variable resistors allowing for the flow of current. (B) The full circuit dia- gram of the membrane as described byHodgkin and Huxley(1952). (C) Equilib- rium functions for the Hodgkin-Huxley gating variables m (Na+_{activation), h (Na}+ inactivation), and n (K+ _{activation). (D) Time constants for the gating variables.} (E-F) Bifurcation diagram and simulations for the Hodgkin-Huxley model. In E, Solid/dashed black lines are stable/unstable steady states, solid green/dotted blue lines are stable/unstable periodic orbits. For low input current I, the neu- ron is at rest (triangle). As I is increased, stable and unstable limit cycles appear through a SNP bifurcation, and a region of bistability exists (enhanced in the inlay), i.e. both resting and spiking regimes coexist (circle). As I is increased further, the steady state becomes unstable via a subcritical Hopf, i.e. the neuron can only spike (square). As I is increase beyond physiological ranges, the limit cycle tends towards low amplitude oscillations (star), which eventually disappear via supercritical Hopf bifurcation (diamond).

more detailed biophysical models of the neuron must be used.

Hodgkin and Huxley (1952) famously studied the squid giant axon to build a biophysical quantitative description of the membrane potential of the neuron. The membrane of the neuron was described as an electrical circuit (shown in Fig- ure 1.12A-B), describing the insulating membrane as a capacitor and ion chan- nels as variable resistors allowing the flow of current. This resistor-capacitor cir- cuit is described by the equation

C ˙V = I − Iion, (1.12)

where I is the applied current across the membrane, Iion is the ionic current, C is the capacitance of the membrane and V is the membrane potential. The ionic currents are described by

IX = gXψX(V − EX), (1.13)

where X is an element in the set of ionic currents (e.g. X ∈ {Na+,K+} for the Hodgkin-Huxley model described below and in Figure 1.5), gX is the maximal conductance of the current X, ψX is the fraction of channels in the conducting state, and EX is the equilibrium potential of the current.

A simple example of a model in this form is the leaky integrate-and-fire neu- ron (Deco et al., 2008), where Iion is an Ohmic ‘leak’ current, i.e. Iion = IL = gL(V − EL). This one-dimensional model can be used to study how proper- ties such as the membrane resistance (R = 1/gL), membrane capacitance (C), and membrane time constant (τ = RC) influence the subthreshold dynamics of neurons in a computationally inexpensive manner. Whilst this model does not simulate an action potential, an AP threshold may be assigned in an integrate- and-fire fashion (section 1.4.2). A crucial advantage of this model is that the frequency-current relationship can be analysed analytically (Deco et al., 2008). Additional biophysical currents of the form Equation 1.13 can be added to the leaky integrate-and-fire model to analyse their roles in modulating the dynamics of a neuron where a detailed knowledge of the mechanisms of action potential generation is not required e.g. the role of the hyperpolarisation activated Ih cur- rent on the generation of subthreshold oscillations (Rotstein et al., 2006) and spatial firing patterns (Bonilla-Quintana et al.,2017).

Hodgkin and Huxley(1952) used this electrical circuit description to model the mechanisms of AP generation. To do so, the activation of sodium and potas- sium currents were quantified using a voltage clamp technique. Their results are described qualitatively in Figure 1.5. Firstly, voltage gated ion channels for the different types of ions were described in terms of gating functions. Namely, Na+ ion channels were modelled by three activation gates (m) and one inactivation

Biophysical modelling of the neuron

gate (h), such that

ψNa = m3h. (1.14)

K+_{ion channels were modelled by four activation gates, so}

ψK= n4. (1.15)

Here, ψX are described inEquation 1.13. Assuming that the proportion of gates which are opening or closing is voltage dependent, each of these gating variables can be modelled as

˙x = αx(V )(1 − x) − βx(V )x, (1.16) where αx(V )and βx(V )are rate constants and x ∈ {m, h, n}. By writing

x∞(V ) = αx(V ) αx(V ) + βx(V ) (1.17) τx(V ) = 1 αx(V ) + βx(V ) , (1.18)

equation 1.16can be rewritten as

˙x = x∞(V ) − x τx(V )

. (1.19)

This equation can be interpreted as follows: x∞(V ) is the voltage dependent steady state value of x, and τx(V )is the voltage dependent time constant for x.

By holding the neuron at a given membrane potential and recording the con- ductance of Na+ _{and K}+ _{currents, equations for these rates of changes of gating} variables could be calculated. Plots of x∞ and τx for each of the gating variables described byHodgkin and Huxley(1952) are shown inFigure 1.12C-D. Under this biophysical formulation of the neuron model, action potentials can be generated which closely match those seen in the data (Hodgkin and Huxley, 1952). This type of ODE formulation makes an analysis of the mechanisms of AP generation possible using dynamical systems theory. Figure 1.12E shows an example bifur- cation diagram for the Hodgkin-Huxley model, demonstrating the bifurcations by which spiking arises as the input current is changed. Simulations of examples of dynamics in the Hodgkin-Huxley model are shown in Figure 1.12F. Biophys- ically realistic models are often more high dimensional and computationally ex- pensive than their phenomenological counterparts, but can be used to study the roles of different ionic currents in modulating neuronal dynamics. In chapter 2, we use a 12-dimensional model in the Hodgkin-Huxley formulation to study the mechanisms underpinning altered temporal firing patterns in an animal model of tauopathy.

mary biological sources of noise in neurons, synaptic noise and channel noise (White et al., 2000). Synaptic noise due to incoming spikes from neighbouring neurons can be modelled as a point Poisson process run through a synaptic re- sponse kernel acting on the membrane potential of the neuron (Fernandez and White,2008). Channel noise arises due to spontaneous fluctuations in the open- ing and closing of ion channels in the neuron and can be modelled in a biophys- ically realistic manner using continuous time Markov chain approaches (White et al.,2000) in which the rate constants αx(V )and βx(V )ofEquation 1.16are de- scribed by a Markov model instead of an ODE. This is computationally expensive, so to reduce computational expense phenomenological SDE approximations of channel noise ranging from white noise currents on the membrane potential to detailed coloured noise on the ion channels may be used (Goldwyn and Shea- Brown,2011).