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Introduction:

In 'Wiring optimization in the brain'(2000), Dmitri Chklovskii and Charles Stevens analyse the dependence of the complexity of cortical circuits on the number of synapses per neuron using virtual perturbations. The authors clarify what they mean by a perturbation in their introduction:

To calculate the optimal wire fraction, we start with a real cortical region containing a fixed number of neurons, a mm cube, for example, and imagine perturbing it by adding or subtracting synapses and the axons and dendrites needed to support them. The rules for perturbing the cortical cube require that the existing circuit connections and function remain intact (except for what may have been removed in the perturbation), that no holes are created, and that all added (or subtracted) synapses are typical of those present; as wire volume is added, the volume of the cube of course increases.

Starting from a reasonable set of physiological relationships approximated as algebraic equations they derive a formula for $\theta$, the ratio of the number of synapses per neuron in the perturbed cortex to that of the real cortex as a function of $\phi$, the volume fraction of the perturbed cortical region made up of wires(synapses + dendrites), and $\lambda$, a constant that multiplies the length of wire associated with each synapse:

\begin{equation} \theta(\phi) = \frac{1}{\lambda^5} \cdot \Big(\frac{1-\phi}{1-\phi_0}\Big)^{\frac{2}{3}} \cdot \frac{\phi}{\phi_0} \tag{1} \end{equation}

In the paper, $\lambda$ is assumed to be equal to $1.0$ and I must say that the authors explain the derivation clearly. It doesn't require more than highschool algebra and each assumption is carefully justified. By calculating $\frac{d\theta}{d\phi}$ you may then find that ,if $\lambda =1$, $\theta$ is maximal when $\phi = 0.6$. This much makes sense to me but the authors then try to rationalise this value in a manner that appears to lack careful justification:

Why does complexity reach a maximum value at a particular wire fraction? When wire and synapses are added, a series of consequences can lead to a runaway situation we call the wiring catastrophe...At this point(phi=0.6), adding wire becomes impossible without decreasing complexity or making other changes - like decreasing axon diameters - that alter cortical function. The physical cause of the catastrophe is a slow growth of conduction velocity and dendritic cable length with diameter combined with the requirement that the conduction times between synapses (and dendrite cable lengths) be unchanged in the perturbed cortex.

Question:

The kind of justification that appears necessary to me concerns data on brain development. I understand that significant synaptic pruning occurs during adolescence so $\theta$ should vary and I think this poses a significant challenge to the account given by Chklovskii and Stevens. More precisely, this variation isn't accounted for by variations in $\phi$ in their model so I have doubts concerning the temporal and causal validity of their model.

To quantify the variation in $\theta$ we may use numbers from 'Synaptic Pruning Mechanisms in Learning' by Deborah Sandoval:

At infancy, each neuron averages around 2,500 synapses and at the peak of exuberant synaptogenesis (around 2-3 years of age) the number increases to around 15,000 (The University of Maine, 2001).

I must add that I have spent some time googling 'Wiring Catastrophe' but this doesn't return anything besides the original paper by D. Chklovskii and C. Stevens.

These observations lead me to the following questions:

  1. If equation (1) is not an exact relationship, how can the extremization(i.e. differentiation) procedure be justified within the context of applied mathematics?
  2. Is there any biological evidence of runaway wiring catastrophes occurring in real brains?

Discussion:

To clarify how synaptic pruning fits into the discussion we may associate a perturbation with a temporal variation in the number of synapses per neuron in the brain as follows:

\begin{equation} \theta(i \rightarrow f) = \frac{\langle \text{synapses}_{t=f}/\text{neuron}_{t=f} \rangle}{\langle \text{synapses}_{t=i}/\text{neuron}_{t=i} \rangle} \approx \frac{\sum \text{synapses}_{t=f}}{\sum \text{synapses}_{t=i}} \tag{2} \end{equation}

since the number of neurons in the brain is approximately constant. My argument is essentially that significant variations in $\theta$ aren't explained by the variations in $\phi$ which stands in contrast to the authors' assertion in their introduction that the relation between relative complexity $\theta$ and wire volume fraction $\phi$ is given by equation (1).

Addendum:

I just received a pretty comprehensive reply from Charles Stevens via email. In particular, he advised me to read the final version of their paper: Chklovskii, D.M., Schikorski, T., and Stevens, C.F. (2002) Wiring optimization in cortical circuits. Neuron 34:341- 347. This is on my reading list for the day.

References:

  1. D. Chklovskii, C. Stevens. Wiring optimization in the brain. NIPS. 2000.
  2. Gal Chechik ,Isaac Meilijson. Neuronal Regulation: A Mechanism For Synaptic Pruning During Brain Maturation. 1999.
  3. Deborah Sandoval. Synaptic Pruning Mechanisms in Learning. 2015.
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  • $\begingroup$ I'm not really clear what you're asking. They're calculating an optimum for the ratio of wire (axons/dendrites) to non-wire (cell bodies). θ is a relative measure so I don't see how adolescence is relevant. The passage you quote as not careful seems perfectly reasonable to me and follows directly from the part you said is straightforward. $\endgroup$ – Bryan Krause Aug 12 at 18:26
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    $\begingroup$ "My argument is essentially that significant variations in θ aren't explained by the variations in ϕ" They are not interested in calculating theta, only maximizing it for a given lambda. They are only interested in the optimal value of ϕ that maximizes theta and showing that it matches what is found in the brain. That optimum does not depend on the other parameters, only on the cable equation assumptions described in the methods. $\endgroup$ – Bryan Krause Aug 12 at 19:02
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    $\begingroup$ @BryanKrause After further reflection I managed to summarise my doubts into a couple questions. $\endgroup$ – user3503 Aug 12 at 19:39
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    $\begingroup$ Ok, I'll swing back later and take a stab at an answer if no one beats me to it. It will likely be mostly a synopsis of these comments though. I think you've still missed the point of the paper a bit and are reading things into it that were never claimed or intended. $\endgroup$ – Bryan Krause Aug 12 at 20:14
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    $\begingroup$ @AidanRocke You can always try to answer your own question. :) $\endgroup$ – Steven Jeuris Aug 13 at 10:51
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The role of synaptic pruning and brain development:

In an email I received from Charles Stevens of the Salk Institute, one of the co-authors, he explained that they were trying to make predictions about cortical structure and function in fully developed brains and that they weren't trying to model brain development. This answers my first question.

 Empirical observations of the 'Wiring Catastrophe':

He also advised me to read the final version of his paper [2] which is a lot more thorough and contains a number of predictions based on three different thought experiments. Each thought experiment leads to a function whose extrema correspond to a scenario where the volume of axons and dendrites occupy approximately 60% of a cubic cortical region.

Instead of referring to a 'Wiring Catastrophe' the final paper simply focuses on the predictions of each thought experiment which are backed by the following observation:

We also measured the wire fraction from three mouse cortical regions and find that its actual value is not significantly different from the special value of 3/5.

To be precise, using an electron microscope they analysed cubic cortical regions from the Neocortex, Piriform cortex and Hippocampus of adult mice.

Finally, I must add that the discussion section and experimental methods section contain very insightful caveats.

References:

  1. D. Chklovskii, C. Stevens. Wiring optimization in the brain. NIPS. 2000.
  2. Chklovskii, D.M., Schikorski, T., and Stevens, C.F. (2002) Wiring optimization in cortical circuits. Neuron 34:341- 347
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