Other global bifurcations involve intersections of homoclinic or heteroclinic orbits either with themselves, or with fixed point attractors. A fascinating example involves the birth of a limit cycle out of a saddle-node bifurcation on a homocline! This obscure-sounding event actually lies at the heart of neuronal firing and we discuss it in more depth in the next section.
Neurons are traditionally seen as the building blocks of the brain. It hence makes sense to gain some insight into their dynamics - and functional interactions - at the microscopic scale at which they reside before moving into the larger scales, which we do in Sect. 4.
The "foundation stone" of microscopic models are the conductance-based Hodgkin-Huxley model and its derivatives. A full description of these is provided by a number of authors (e.g. Izhikevich 2005, Gerstner & Kistler 2002, Guevara 2003). Our objective here will be to quickly move from the full model to a two dimensional approximation and then explicate the onset of neuronal firing as a dynamical bifurcation.
Before we do this, for the sake of theoretical and historical completeness, we briefly discuss the McCulloch-Pitts model (1943), z(xi,t+ 1) = S E hijz(xj,t) - £j I , (30)
y where hij is the connectivity matrix, e the "threshold" of neuron i and S is the step function. Neural inputs to a given unit are summed and then converted into a binary output if they exceed the threshold £j. The resulting output is iteratively fed back into the network. Hence the McCulloch-Pitts model is discrete in both space and time, and as such is an example of a coupled difference map (5). Considered together with the use of the step function as representing neural "activation", this model is perhaps as abstract as possible. Nonetheless, McCulloch and Pitts proved that the system was capable of remarkably general computational feats. However, it is probably fair to say that this model finds its place more appropriately in the lineage of artificial neural networks than in the understanding of the dynamics of biological systems. Hence, McCulloch-Pitts systems form the basis for the two-layer "per-ceptrons" of Rosenblatt (1958) and the symmetric Hopfield (1982) networks. These extend the complexity and computational properties of McCulloch-Pitts systems to permit object categorization, content-addressable memory (i.e. the system correctly yields an entire memory from any subpart of sufficient size) and learning. For example, Sejnowski and Rosenberg (1987) showed that such systems, if constructed with three interconnected layers, are able to learn language pronunciation.
An overview of related advances is provided by Ermentrout (1998). For a fascinating history of this model and the life of Walter Pitts, see Smalheiser (2000).
3.2 Biophysical Models of the Neuron: The Hodgkin-Huxley Model
Whereas the McCulloch-Pitts system was constructed to embody only the very general network properties of neural systems and to directly address computational issues, the Hodgkin Huxley model aims to incorporate the principal neurobiological properties of a neuron in order to understand phenomena such as the action potential. Computational properties of these neurons are then investigated.
The paper of Hodgkin and Huxley (1952) is remarkable in that it casts detailed empirical investigations of the physiological properties of the squid axon into a dynamical systems framework. The Hodgkin-Huxley model is a set of conductance-based coupled ordinary differential equations6 of the form of equation (8), incorporating sodium (Na), potassium (K) and chloride ion flows through their respective channels. Chloride channel conductances are static (not voltage dependent) and hence referred to as leaky (L). Hence we have,
= gNafNa (V (t)) X (V (t) — VNa) + 9K Ik (V (t)) X (V (t) — Vk ) + gL X (V (t) — Vl)+ I, (31)
where c =1 ^F/cm2 is the membrane capacitance, I is an applied transmembrane current and Vion are the respective Nernst potentials. The
6 Here we depart slightly from the traditional nomenclature in order to simplify the mathematical description of the model.
coefficients gion are the maximum ion flows in the case where all the channels of that ion species are open. The Na and K ion flows reflect the state of "activation" channels, which open as membrane voltage increases and "inac-tivation" channels, which close. These are given by, fNa(V ) = m(V )M h(V )H
where m and n are activation channels for Na and K, and h is the single inactivation channel for Na. The exponents are determined by the number of such classes of channel M = 3, H =1 and N = 4. Hence (31) reflects the combined flow of all ion species as they are "pushed through" open channels according to the gradient between the membrane and Nernst potentials.
The kinetics of activation and inactivation channels are determined by differential equations of the form, dm(V) = (mx (V) - m (V))
where mTO(V) is the fraction of channels open if the voltage is kept constant and Tm(V) is a rate constant. These are determined empirically. These equations embody the exponential relaxation of channels towards their (voltage-dependent) steady states mTO(V) consequent to a transient change in membrane potential. The kinetics of h and n are of the same form, although their rate constants t are obviously distinct. The form of mTO(V) - the steady state configurations of ion channel populations as a function of membrane potentials - is sigmoid shaped of the form, m~(V)=1 + mTv)/„ • (34)
where Vm is the threshold potential for the ion channel and a introduces the variance of this threshold. Figure 14 summarizes membrane dynamics in the Hodgkin-Huxley model.
The Hodgkin-Huxley model is a conductance approach to the dynamics of neural activity, reflecting ion flows through voltage- and time-dependent transmembrane channels. It represents a beautiful juncture of empirical and mathematical analysis. It not only offers an explanation of neural firing, but it quantitatively captures the complex shape of a neural depolarization.
3.3 Dimension Reductions of the Hodgkin-Huxley Model
The Hodgkin-Huxley model is able to explain the chief properties and many of the nuances of neuronal depolarization, including the threshold effect and the post-depolarization refractory period with quantitative accuracy. However, much of the qualitative (and some of the quantitative) behaviour can be
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