# Excitability of neurons

[The following includes a) two specific questions (at the end), b) an attempt to capture a dispositional concept (excitability) in geometric and physical terms.]

I assume that "excitability of a neuron" is a reasonable concept (and measure) to distinguish neurons: there are (as will be seen) neurons that are more or less excitable (in a quantifiable way).

What does "excitability" mean?

Excitability $E$ may be operationally defined as the sensitivity of a neuron to excitatory synaptical inputs. Mathematically stated: as the (decimal logarithm of the reciprocal of the) proportion of excitatory synapses that must be simultaneously active (i.e. generate an excitatory postsynaptic potential, EPSP) in order to collectively evoke an action potential.

Let $N$ be the number of all excitatory synapses of the neuron, $n$ be the the minimal number of simultaneously active synapses that is needed to evoke an action potential:

$$E = \log \frac{N}{n}$$

If the proportion is 100% (all excitatory synapses must be active), $E = \log (1) = 0$, if it's only 10%, $E=1$ (higher excitability), if it's 1%, $E = 2$, and so on.

How to determine $E$ for a given neuron? With the operational definition given above:

1. Count the actual number $N$ of excitatory synapses.

2. Estimate the minimal number $n$ of simultaneously active synapses that is needed to evoke an action potential (by repeated experiments).

Especially the second number $n$ will be quite hard to achieve experimentally. But there is an alternative way to estimate this number, given a simplified model of the neuron:

• All synapses have equal (mean) distance $r$ to the axon hillock (where the action potential is generated).
• The EPSP generated at a synapse is $u$.
• The EPSPs travelling from the synapse to the axon hillock have a constant decay constant $\alpha$.
• The threshold value for action potential generation is $\theta$.

When $m$ synapses simultaneously generate an EPSP of size $u$, these EPSPs will sum up at the axon hillock to

$$U = m\times u\times 10^{-\alpha r}$$

Only if $U \geq \theta$ an action potential will be generated, that is, the minimal number of simultaneously active synapses to generate an action potential is

$$n = 10^{\alpha r}\ \frac{\theta}{u}$$

With this we can estimate the excitability $E = \log\frac{N}{n} = \log 10^{-\alpha r}\frac{N\ u}{\theta}$ by

$$E = -\alpha r + \log(N) + \log \frac{u}{\theta}$$

$\alpha$, $u$ and $\theta$ are more physical constants (more or less the same for all types of neurons?), $r$ and $N$ depend heavily on the arbitrary geometry/morphology of the neuron (especially do longer and more strongly branched dendrites have more synapses).

My specific questions are (please feel free to answer only with "yes" or "no"):

1. Has the property of excitability (in the operational sense above) been investigated a) for single neurons, b) for morphological types of neurons?

2. Is there an observed significant correlation between morphological type and excitability? Note that larger dendritic trees have larger $r$ (decreasing $E$) but also larger $N$ (increasing $E$).

(Final remark: The relation between $r$ and $N$ of a neuron is partially reflected in the structure – especially the degree of branching – of its dendritic tree.)

Excitability has definitely been tested and measured, but I'm not sure about your exact definition (sort of hard to prove a negative).

For differences in "excitability" across neuron types, I can think of two general approaches.

1) You could simply consider the firing rate and define cells as more "excitable" simply if they fire more or have a lower current threshold (though one must be careful in interpreting the actual "mV" value of threshold, because resting potential and input resistance are crucial and thresholds are dynamic).

These parameters have nothing to do with the number of synapses on them or the number of synapses it takes to fire; if you focus too much on these parameters you risk making a comparison that isn't all that helpful (see the analogy below). This approach is most suited to comparisons across cell types.

2) The second type of excitability concerns the intrinsic properties of a cell: what is it's input resistance, spike threshold, how much current do you need to inject at the soma to evoke a spike, etc. These properties can indeed vary across cell types, but you really have to think about it in the context of (1) to avoid the same problems with the analogy below. A cell might be near threshold but receives very small inputs; should that cell be considered more excitable than another?

Instead, it makes more sense to think about changes of excitability: "how does Manipulation X affect the excitability of Cell K?" If it changes the amount of current you need to inject to the soma to evoke a spike by depolarizing the cell or increasing the input resistance, you would say it increases excitability. This is, in my experience, by far the primary way in which "excitability" is discussed in neuroscience, rather than as a factor that varies across neurons. I did a search on Google Scholar for "neuron excitability": every one of the first 20 results was using it in the context of a change in excitability from one condition or manipulation to another.

A tangential analogy (the problem with your proposed definition of excitability):

Your proposed definition of excitability is a bit like measuring income based on paycheck frequency or number of digits on the check. You could certainly do comparisons between jobs on those merits, and in some cases they might be valid for what you'd normally think of as "income", but as soon as you move between countries (i.e., brain regions or cell types) currency changes, and suddenly the number of digits doesn't help because you don't know if you are talking about British pounds, US dollars, or Brazilian real. And even within a population, if the paychecks come weekly rather than monthly, the same face value of a check means a totally different annual income (i.e., if a cell has synapses that are rarely active, they contribute less to depolarizing it than synapses that are active all the time).

Contributions to excitability

Your question proposes that morphology is a defining characteristic of excitability. In a previous question you asked, I directed you to a review by Magee, J. C. (2000). Dendritic integration of excitatory synaptic input. Nature reviews. Neuroscience, 1(3), 181. - this review is applicable here, too. Active conductances in the dendrites and variation of synaptic properties with distance from the soma (larger, faster EPSPs the further from the soma you go) act to regularize the influence of EPSPs with distance, so you cannot rely only on passive properties. That also means you can't simply assume that a cell with lots of synapses far from the soma is not excitable.

However, there are definitely differences in firing rates and firing thresholds across neuronal subtypes, and you can take these to mean differences in excitability. There are many papers that talk about the diversity of intrinsic properties or firing rates of cells. I'll list a couple here for further reading:

Llinás, R. R. (1988). The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function. Science, 242(4886), 1654-1664.

Izhikevich, E. M., & Edelman, G. M. (2008). Large-scale model of mammalian thalamocortical systems. Proceedings of the national academy of sciences, 105(9), 3593-3598.

It's also important to recognize that excitability is dynamic and cells that are highly excitable in one state may not be as excitable in another state, even though their morphology does not change.

• I am glad you mention Llinas: I just started reading his I of a vortex. – Hans-Peter Stricker Sep 8 '17 at 20:20
• Thanks for your demanding answer, anyway. I'll try to make the best out of it. – Hans-Peter Stricker Sep 8 '17 at 20:24
• Careful with Llinas, I link to him with caution. He's a good scientist and easy to read, but in his reviews and books he is definitely writing from a persuasive angle. I don't mean that in some nefarious way, but he definitely has an opinion in mind that he's trying to convince you about; not everyone in the field would agree with everything he tries to convey. – Bryan Krause Sep 8 '17 at 20:25
• I am careful, and know what you mean - but he is not the only one. Do you know of one prominent author to which your precautions don't apply? – Hans-Peter Stricker Sep 8 '17 at 20:28
• Getting off topic here, but I think the distinction is more of writing style; some write like they are telling you the truth and don't really give away that there is another way. Such writing is often the most compelling. Others write intending to relate the state of the field, including the controversy, and make their opinions more obviously opinions. Of course caution is always warranted, but I mean to suggest that, at least in my experience and opinion, Llinas likes to start with the premise that his thinking is correct without much service to the possibility he is not. – Bryan Krause Sep 8 '17 at 20:34