Brains as spiking attractor networks

Thank you to everyone who commented, online and in person, on my previous post. I hope this updated version will address the issues you raised, but I warmly welcome further comments. If the following is good enough I'll use it in a video entitled 'Basic neuroscience' that I'm making for YouTube, and in my analysis of neuronal multielectrode array data (Harris et al., 2010).

What I'm after is agreement on a simple but formally correct quantitative framework for describing brain activity. What determines the spiking (1) or non-spiking (0) of a neuron at a given time t is, as several people pointed out, an enormously complex interplay of cellular and synaptic processes, and it is the job of neuroscientists to discover the conditions (e.g. synaptic weights) necessary for artificial neural networks to replicate the spike patterns of biological brains. However, the spiking/non-spiking binary is clearly the most salient computational property of neurons, and in multielectrode recordings it is often the only value we can reliably detect.

Several people pointed out that the state of a brain (X) at time t, is a vector (an array of values) rather than a sum, i.e.

X(t) = [V₁(t), V₂(t),... V_N(t)]

where N is the number of neurons in the brain and V is the spiking (1) or non-spiking (0) of a particular neuron. Thus, a brain containing 100 neurons has 2¹⁰⁰ possible states, e.g.

X(t₁) = [ 0 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 ]^T

X(t₂) = [ 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 ]^T

X(t₃) = [ 0 0 0 0 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 ]^T

where red highlights a change in neuronal activity from the previous time point, and T stands for transpose (change from row to column) since the states of a neuron at successive time steps are usually arranged in a row (e.g. a raster plot, spike density function or voltage trace).

So far so good, but now to the heart of the matter: how do we use this simple framework to describe the complex brain activity manifest in cognition, behaviour and key neural processes?

The 2¹⁰⁰ (1.2676506 × 10³⁰) possible states of a brain with 100 neurons are referred to as its state space. Some of these states are not possible under biologically plausible conditions, and only a fraction of all biologically plausible states will be expressed during the lifetime of a brain. The order in which brain states occur is referred to as the trajectory of the brain's activity in state space. If we plot against one another the spike patterns (e.g. spike density functions) of two or more antagonistic neurons or neural networks, such as those that drive walking, breathing or swimming, they form a cyclic trajectory in state space, called an attractor, e.g.

Cyclic attractors in state space. After displacement (blue) one system (a) returns to its main trajectory while a different system (b) switches to a different trajectory, which drives a different behaviour. Figure from Briggman and Kristan (2008) Multi-Functional Pattern Generating Circuits.

Some commenters questioned the usefulness of the attractor concept for understanding complex cognitive processes and behaviours, such as making coffee, that involve large brain networks. In the absence of an alternative though, I still think the attractor model is worth pursuing, because the activity of sensory and motor neurons is tightly constrained by complex behaviours, and can therefore be described as attractors in the state space of the brain. In making coffee for instance, extending the arm and grasping the kettle must precede pouring hot water into the cup, and all three actions map directly onto tuning curves of specific neurons in the hand and arm regions of the left (typically) motor cortex. Therefore, these neurons will generate very similar spike patterns every time the action is performed. I believe the activity of neurons not directly constrained by the sensory and motor demands of the task, and therefore not obviously part of the behaviour, will nevertheless be selected for its ability, through direct and indirect synaptic connections, to create and maintain the required attractor among the sensory and motor neurons.

In terms of the present framework the question is, for each neuron or group of neurons N, how similar their spike patterns V have to be, morning-to-morning, for coffee to be successfully made. Neurons that are constrained in some way can be considered part of the coffee-making attractor. We can approach the question for instance by hypothesizing that the activity V of neurons in the pre-motor cortex will be more constrained than that of neurons in the temporal cortex, and then use EEG to measure variability in these regions on successive mornings. In animal models the recordings could be at the level of individual neurons and perturbations could be introduced. The relationship of the spike pattern of a neuron to an ongoing attractor may be complex and depend on the spike patterns of many other neurons, and some behaviours clearly require more neuronal resources than others, but these are empirical questions that, to me at least, seem fruitful.

Some further questions:

• In what ways can some neuronal activity V_N(t) be said to participate in an ongoing attractor?
• How do we characterize, mathematically and visually (or perhaps audibly) an attractor in N dimensional space?

Lorenz attractor