Kinetic Models of Brain Activity

Abstract

Brain imaging sciences, like neurosciences in general, have predominantly been an empirical endeavour. This paper argues that the maturation of “kinetic models” of large-scale neuronal activity will provide a unifying theory to underpin brain imaging sciences. In particular, this framework will provide a means of unifying data from different imaging modalities, afford a direct link with cognitive theories of brain function, equip researchers with novel data analysis methodologies and underpin a dialogue in which theoretical formalisms are iteratively refined or refuted through empirical studies. Three steps are crucial to this endeavour: (1) The extension of models of spiking neural ensembles (where the states of all neurons are specified) to statistical models of neural “masses” (where only a few moments of the distribution of states are specified); (2) The refinement of “forward models”, such as neurovascular coupling, which map neuronal states to observables; and (3) A theory which links the distribution of neuronal states to cognitive operations, hence informing cognitive neuroscience experiments. We provide illustrative examples of kinetic models of neuronal dynamics at the mesoscopic scale, focusing on the manner by which sensory inputs modify the expression of ongoing “background” activity. The paper concludes with an overview of some of the cutting edge developments in kinetic models of brain activity.

Appendices

Appendix A: Biophysical models of spiking neurons

Biophysical models of spiking neurons derive from the Hodgkin Huxley framework. An elegant description of their dynamical principles is provided in Izhikevich (2005). A recent summary account is Breakspear and Jirsa (2007). The approach here is to provide plausible examples of dynamics within large networks of spiking neurons.

These models are based on ion currents through leaky and voltage-dependent transmembrane channels and have the form,

$$C\frac{{{\text{d}}V}}{{{\text{d}}t}} = I + {\sum\limits_{{\text{ion}}} {f_{{{\text{ion}}}} {\left( V \right)} \times {\left( {V - V_{{{\text{ion}}}} } \right)}} }$$

(3)

where V is the membrane potential, ion denotes various ion species (Na⁺, Ca²⁺, K⁺), f _ion are the voltage-dependent conductances for each ion species and I is a current, either experimentally injected (as per in vivo preparations) or induced at the synapse by ligand-gated channels. The functions f _ion represent the states of voltage-gated activation and deactivation channels, generating currents due to the gradient between the membrane potential V and the Nernst potential for that ion species V _ion. For the present purposes, we incorporate only voltage-dependent Na⁺ and K⁺ activation channels,

$$f_{{\text{Na}}} \left( V \right) = g_{{\text{Na}}} .m\left( V \right),$$

(4)

$$f_{\text{K}} \left( V \right) = g_{\text{K}} .n\left( V \right).$$

(5)

The functions m and n are the voltage-dependent fraction of open channels of each respective ion species. These are then multiplied by the conductances g _ion to give the ion currents. Sodium channels respond very quickly to changes in membrane potential, whereas potassium channels relax towards their steady state values relatively slowly. Hence,

$$m\left( V \right) = \frac{{m_{\max } }}{{1 + e^{\left( {{{V_m - V} \mathord{\left/ {\vphantom {{V_m - V} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} \right)} }}.$$

(6)

represents the instantaneous state of sodium channels, where m _max is the maximum fraction of open sodium channels, V _m the voltage at which half the channels are open, and σ controls the range of voltages over which the change occurs. On the other hand,

$$\frac{{{\text{d}}n\left( V \right)}}{{{\text{d}}t}} = \frac{{\left( {n_\infty \left( V \right) - n\left( V \right)} \right)}}{{\tau _n \left( V \right)}}$$

(7)

with,

$$n_\infty \left( V \right) = \frac{{n_{\max } }}{{1 + e^{\left( {{{V_m - V} \mathord{\left/ {\vphantom {{V_m - V} \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} \right)} }}.$$

(8)

captures the relaxation of potassium conductance towards its voltage-driven state, with τ _n the time constant of the potassium channel relaxation, and n _max, V _n , and σ have the same meaning as the sodium channels. As discussed in Izhikevich (2005), such a model is the minimum required in order to exhibit both type I and type II neuronal firing. Type I neurons (integrators) fire through a saddle node bifurcation and, because of this, act to integrate the average rate of synaptic input, firing at a proportional rate if a threshold is surpassed. Type II neurons (resonators) fire through a Hopf bifurcation and, hence, show a resonance to a preferred frequency. The influences of other neurons, nonspecific noise and sensory-evoked synaptic activity are introduced via the synaptic current term. Hence, if we take an ensemble of N neurons {1,2,…, N}, then the synaptic current at neuron j is due to three inputs,

$$I^j = I_{{\text{ext}}}^j + I_{{\text{noise}}}^j + \sum\limits_k {H_c \left( {g\left( {V^k } \right)} \right)} .$$

(9)

I _ext represents a synaptic current induced by specific inputs from outside the local ensemble—such as due to a sensory stimulus relayed via the thalamus. I _noise represents a non-specific stochastic input—representing all subcortical and cortico-cortical inputs not accounted for by I _ext plus local stochastic influences such as amplified channel noise (Faisal et al. 2008). The final term represents inputs summed from all other neurons within the ensemble. The coupling function H is a connectivity map that introduces the presence and weighted strength of synaptic connections between neurons j and k. The function g incorporates two processes. The first is a thresholded step function that yields a brief burst of activity (a boxcar function) whenever neuron k issues an action potential. The second process is the conversion of this synaptic activity into a post-synaptic current at the index neuron j. For the present purpose, the post-synaptic current does not have an intrinsic time constant. Hence the frequency response of the neuron is determined by the membrane time constants. More interesting synaptic forms, which act as temporal filters, introduce additional time scales into such models (Fourcaud and Brunel 2002).

The parameters of the individual neurons were set in a physiologically realistic range and so that, in the absence of any synaptic current I = 0 each neuron is quiescent. In response to a sufficiently strong stochastic input, neurons fire stochastically due to intermittent subthreshold excursions. However, in the presence of a sufficiently strong external current I _external each neuron undergoes a “bifurcation” from its quiescent state to spontaneous and periodic firing. This bifurcation is known as a “saddle-node” bifurcation (see Izhikevich 2005; Breakspear and Jirsa 2007).

The parameters are:

Capacitance:

C = 1;

Synaptic current (default):

I = 0;

Leaky channels:

V _L = −80; g _L = 8;

Sodium channels:

V _Na = 60; g _Na = 20; V _m = −20; σ _m = 15; m _max = 1; τ _m = 1;

Potassium channels:

V _K = −90; g _K = 10; V _n  = −25; σ _n  = 5; n _max = 1; τ _n  = 1;

Appendix B: Neural mass of cortical columns

Neural mass models aim to reduce the computational load of large networks of spiking neurons (Freeman 1975). The equations here begin with a model of the behavior of local ensembles of neurons, with dynamical variables V = [V W Z] representing ensemble averages over the extent of a local neural region (Eqs. 10, 11, 12, 13 and 14). The effect of the many thousands of inputs (i.e. the local mean field) into each local subsystem from other neurons is then introduced by coupling such nodes together through long-range excitatory projections (Eqs. 15 and 16) The model here was extended from a description by (Larter and Speelman 1999) in Breakspear et al. (2003a). An alternative neural mass model is the Jansen model (Jansen and Rit 1995) as recently further elaborated by Garrido et al. (2007). As with the spiking neural ensembles, the present goal of these neural mass models is to provide plausible examples of interesting and computationally relevant dynamics.

The dynamical variables represent the mean membrane potential of pyramidal cells, V, and inhibitory interneurons, Z, and the average number of ‘open’ potassium ion channels, W. The evolution equations are adapted from a study of epileptic seizures in hippocampal slices (Larter and Speelman 1999), which in turn are derived from the model of Morris and Lecar (1981) by introducing feedback and population effects. The mean cell membrane potential of the pyramidal cells is governed by the conductance of sodium, potassium and calcium ions through voltage- and ligand-gated membrane channels,

$$\frac{{{\text{d}}V}}{{{\text{d}}t}} = - \left( {g_{{\text{Ca}}} + r_{{\text{NMDA}}} a_{{\text{ee}}} Q_{\text{V}} } \right)m_{{\text{Ca}}} \left( {V - V_{{\text{Ca}}} } \right) - \left( {g_{{\text{na}}} m_{{\text{na}}} + a_{{\text{ee}}} Q_{\text{V}} } \right)\left( {V - V_{{\text{na}}} } \right) - g_{\text{K}} W\left( {V - V_{\text{K}} } \right) - g_{\text{L}} \left( {V - V_{\text{L}} } \right) + a_{{\text{ie}}} ZQ_{\text{Z}} + a_{{\text{ne}}} I_\delta ,$$

(10)

$$\frac{{{\text{d}}Z}}{{{\text{d}}t}} = b\left( {a_{{\text{ni}}} I_\delta + a_{{\text{ei}}} VQ_{\text{V}} } \right),$$

(11)

where g _ion is the maximum conductance of each population of ion species if all channels are open, m _ion is the fraction of channels open, V _ion is the Nernst potential for that ion species, a _ab are synaptic weights between respective populations and Q _V(Z) is the firing rate of the excitatory (inhibitory) neurons. The fraction of open channels is determined by the sigmoid-shaped ‘neural activation function’,

$$m_{{\text{ion}}} = 0.5\left( {1 + {\text{tanh}}\left( {\frac{{V - V_{\text{T}} }}{{\delta _{{\text{ion}}} }}} \right)} \right),$$

(12)

where δ _ion incorporates the variance of this distribution. The fraction of open potassium channels is slightly more complicated, being governed by W, with

$$\frac{{{\text{d}}W}}{{{\text{d}}t}} = \frac{{\phi \left( {m_k - W} \right)}}{\tau },$$

(13)

where ϕ is a temperature scaling factor and τ is a ‘relaxation’ time constant. Cell firing rate is also determined by sigmoid activation functions,

$$Q_{\text{V}} = 0.5xQ_{V_{\max } } \left( {1 + \tanh \left( {\frac{{V - V_{\text{T}} }}{{\delta _{\text{V}} }}} \right)} \right),$$

(14)

where the Q _max is the maximum firing rate. An analogous term is also introduced for the inhibitory cells. The firing of these cell populations feeds back onto the ensemble through synaptic coupling to open ligand-gated channels and raise or lower the membrane potential accordingly. In the case of excitatory-to-inhibitory and inhibitory-to-excitatory connections, this is modelled as additional inputs to the flow of ions across the membrane channel, weighted by functional synaptic factors, a _ei and a _ie. In the case of excitatory to excitatory connections, the rate of firing Q _v is assumed to lead to a proportional release of glutamate neurotransmitter across the synapse, onto two classes of ligand-gated ion channels: (1) AMPA channels, which open an additional population of sodium channels, and (2) NMDA receptors, which open an additional population of voltage-gated sodium and calcium channels. r _NMDA incorporates the ratio of NMDA to AMPA receptors. Finally, a stochastic synaptic current term, $I_\delta$ is also present and weighted by the synaptic efficacy term a _ne.

Each of the set of Eqs. 10, 11, 12, 13 and 14 govern the dynamics within each local cell assembly, which hence require inputs from other cortical sources. Coupling between N nodes is introduced as competitive agonist excitatory action at the same populations of NMDA and AMPA receptors. Locating the i-th node at position x _i this is represented as,

$$\begin{array}{*{20}l} {{\frac{{dV{\left( {x_{i} ,t} \right)}}}{{dt}} = F_{{\text{a}}} {\left( {V{\left( {x_{i} ,t} \right)} + c{\sum\limits_j {H_{{ij}} V{\left( {x_{j} ,t} \right)}} }} \right)},} \hfill} & {{i = 1,2, \ldots ,N} \hfill} \\ \end{array}$$

(15)

for i,j = 1,…,N. The function F represents the within-node dynamics as given by B1 and H _ij is the connection strength between nodes i and j. The parameter c represents the strength of excitatory coupling between cortical columns. If c = 0 the systems evolve independently. c > 0 introduces interdependences between consecutive columns. c = 1 corresponds to maximum coupling, with excitatory input from outside each column surpassing excitatory input from within each column.

The spatial kernel H _ij can hence be viewed as a ‘synaptic footprint’—the dependence of coupling strength on distance. In the present simulations we employ,

$$\begin{array}{*{20}c} {H_{{ij}} = \frac{1}{{{\left| {x_{i} - x_{j} } \right|}^{{\mathbf{K}}} }},}{j \ne i,} \\ { = - 1,}{j = i,} \\ \end{array} $$

(16)

K > 0 is a constant vector which parameterises the shape of the synaptic footprint. When K is close to zero, H _ij is almost constant, so that ‘mean field’ effects dominate. Increasing K decreases the effective support of H _ij towards a neighborhood of x _i . Note that K is of the same dimension of the modelled array, in order to permit anisotropic coupling K ₁ K ₂.

All physiologically measurable parameters (conductances, threshold potentials and Nernst potentials) are set to their accepted values (Larter and Speelman 1999). The behavior of this model is more fully described in Breakspear et al. (2003a).