The cell-attached soundtrack of calcium imaging

Old-school electrophysiologists like to listen to the ephys signals during experiments. For example, this allows to precisely hear when the patch pipette approaches a target neuron. The technique is discussed in the Axon Guide: “Audio Monitor: Friend or Foe?”.

The following is something very similar: Calcium imaging of neuronal activity (GCaMP6f), and an audio track that is derived from a simultaneously performed cell-attached recording of the same neuron (detected action potentials are convolved with a particular sound event, like two metals hitting each other, the sound of an anvil, and the sound of gunshots – the “firing” neuron).

Three times the same calcium recording, but with different soundtracks:

CC-BY 3.0, http://soundbible.com/1750-Hitting-Metal.html

CC-BY 3.0, http://soundbible.com/1742-Anvil-Impact-1x.html

CC-BY 3.0, http://soundbible.com/2123-40-Smith-Wesson-8x.html

Posted in Calcium Imaging, electrophysiology, Imaging, Neuronal activity, zebrafish | Tagged , , , | Leave a comment

Post-publication review: Somato-dendritic coupling of L5 neurons in V1

It requires more than a quick look at the abstract and the figures to fully understand a research paper and its limitations. One way to get there is to write a summary or critical review of a paper. In a contribution to (informal) post-publication review, I will select neuroscience papers that are, in my opinion, worth the time needed to write up a review. other reviews: On precise balance in the hippocampus [1]; on simulations of calcium imaging datasets [3].

Selected papers:

1. Beaulieu-Laroche, Harnett et al., Widespread and Highly Correlated Somato-dendriticActivity in Cortical Layer 5 Neurons, Neuron (2019)

2. Francioni and Rochefort, High somato-dendritic coupling ofV1layer 5 neurons independent of behavioural state and visual stimulation, bioRxiv (2019)

L5 neuron

Simplified scheme of a L5 pyramidal neuron. From Wikipedia, used under CC BY 4.0 license (modified).

What are the papers about? The mammalian cortex is a layered structure that harbors different neuronal types in different layers. One of the most prevalent and most fascinating neuronal types is the layer 5 (L5) pyramidal neuron (schematic drawing above). While the cell body resides deep in layer 5, several hundred micrometers below the surface, it sends a so-called ‘apical trunk’ dendrite up to superficial layers, where it branches into a so-called ‘apical tuft’. The apical tuft receives input which is fundamentally different from input received at the soma. This, together with the long distance between apical tuft and soma, poses a simple, but difficult-to-answer question: how tightly are electrical activity at the soma, the apical trunk and the apical tuft coupled, and how do the electrical compartments interact? The two papers address exactly this question. – Their main finding is that calcium activity at the soma and the apical dendrite are very tightly coupled, no matter what happens to the mouse.

Why this question is important: Six months ago, I wrote a blog post which explained why it would be so interesting to see uncoupled activity of soma and the apical dendrite. Several studies reported – using indirect methods – that there was strong uncoupling happening during sleep or motor learning, but the few studies working on this topic that used direct methods have rather shown tight coupling between apical and somatic activity (Helmchen et al., 1999; Kerlin et al., 2018), with only few outlier events.

These two papers try to address this question again, in a different brain region, in various behavioral conditions, and probably more systematically and in a more targeted way than previous studies.

The method: Both teams performed simultaneous calcium imaging in two optical layers from L5 neurons in the primary visual cortex (V1) of transgenic GCaMP6 mice. The two layers covered either soma and apical trunk, or apical trunk and apical tuft.

Calcium does only partially reflect the electric events that happen in the respective location. Calcium events are an imperfect, non-linear and supra-threshold mirror of electrical activity, and, in addition, there are calcium events independent of electrical activity. Still, calcium imaging is the best available method to investigate somato-dendritic coupling, since theoretically better-suited methods (dendritic patching, voltage imaging) do not work reliably in awake animals for now.

I also would like to highlight the section “Limitations of the use of calcium imaging to assess dendritic activity in awake behaving mice” from the Francioni and Rochefort paper, which discusses in a bit more detail the challenges for dendritic calcium imaging. I think it is a very good idea to include this part in the main paper and not in the methods section, because this is an experiment that requires careful analysis due to motion artifacts, unwanted signal from spines or axonal boutons, imperfect calcium indicators and other problems.

Paper 1 (Harnett lab): The main result of the Beaulieu-Laroche et al. paper is that somatic activity and activity measured in the distal apical trunk (distance between dendritic and somatic regions: ca. 350 μm) are strongly correlated in V1. One would expect that this tight coupling becomes loose when the dendritic tree is bombarded by more input. Surprisingly, however, they find that this tight coupling is independent of visual stimulation or whether the mouse is running or sitting still.

I haven’t talked to the authors, but from this exploratory study design, which tested different conditions (visual input vs. no black screen, running vs. pausing), I would guess that they were hoping to see a behavioral regime where the tight somato-dendritic coupling breaks down, due to strong excitatory apical inputs, due to a modulation of the leakiness of the apical trunk, due to shunting inhibition or another weird phenomenon on the molecular level. The fact that nothing of this happens seems a bit disappointing from the perspective of the study director.

To better understand these calcium imaging measurements, the authors performed calibration experiments in vitro (slices), where they measured both calcium signals in somata and dendrites and patched the respective locations with micropipettes to record the electrical signals. Somehow, the whole-cell recordings did not wash out the GCaMP from the neurons (Fig. S3), which was a bit surprising to me. Using these slice experiments, they conclude that the high somato-dendritic correlations are either due to burst-firing somata that trigger (active) dendritic electrogenesis, or due to dendritic spikes that elicit somatic firing. The authors are unfortunately unable to discern these two possibilities.

To make things a bit more complex, a side-figure shows that single L5 spikes are not seen using the GCaMP6f reporter (but are seen using an OGB-1 calcium dye, see Fig. S3G,H). The authors find that single somatic L5 spikes also do not trigger dendritic electrogenesis, which seems to explain that the calcium signal is correlated also in this regime of single spikes. Indirectly, this suggests that there is no tight coupling of calcium activity between soma and apical dendrite if there is only one somatic spike.

Paper 2 (Rochefort lab): Francioni and Rochefort perform similar calcium imaging experiments. Due to technical reasons (using a piezo instead of a tunable lens), their two imaging planes are closer to each other (<200 μm). As a main result, the authors come to the conclusion that calcium activity is highly correlated not only between soma and (proximal) apical trunk, but also between the proximal and distal trunk and between distal trunk and the apical tuft; and also between processes within a single apical tuft. Including not only the apical trunk but also the apical tuft is an important extension compared to the paper from the Harnett lab. As another difference, they use for some of their experiments GCaMP6s instead of GCaMP6f to have a more sensitive readout of somatic activity.

Very similar to Beaulieu-Laroche et al., Francioni and Rochefort report that this tight somato-dendritic coupling is not affected by visual stimulation or by locomotion, also for the apical tuft. Actually, they report that “less than 3% of the total number of transients” were branch-specific, and that those few events were mostly low-amplitude signals.

The authors then analyze the data more carefully and find that quite some calcium events that can be seen in the soma do not propagate into the apical trunk, and many events in the apical trunk do not propagate into the apical tuft. They quantify the overall loss from somatic activity to the dendritic tuft to ca. 40%. In my opinion, this suggests that most of the calcium activity is generated first in the soma and then propagates into the apical compartment, where it is (according to Beaulieu-Laroche et al.) amplified by active conductances, but not always (as seen by Francioni and Rochefort). One remaining question is why the attenuation from soma to apical trunk has not also been observed by Beaulieu-Laroche et al., but this is probably due to technical reasons, for example the calcium indicator.

Consequences for understanding the role of apical dendrites: During the last 20 years, many people, in particular neuroscientists coming from slice physiology or from theory, have speculated about active roles of dendrites and the computational independence of dendritic segments. Although these two studies (hopefully) are only the beginning of more detailed research covering other brain regions and other behavioral conditions with similar methodological approaches, the findings seem to dampen the enthusiasm about dendritic processing a bit.

Francioni and Rochefort make it very clear, that “this result is consistent with two possible mechanisms: either (1) global apical tuft activation systematically triggers somatic action potentials, or (2) global tuft calcium transients are triggered by back-propagating action potentials alone or in conjunction with tuft synaptic inputs.”

Both options would be possible. In my opinion, however, their paper points rather towards the second option. First, they see that 40% of somatic events do not make it to the apical tuft; these are very likely back-propagating action potentials that get stuck in the middle of their journey in the apical dendrite. Second, given that they do not see the other phenomenon prominently (dendritic events that do not make it to the soma, <3% of events), it seems very likely that a very large fraction of all dendritic events are evoked by back-propagating action potentials.

This suggests a (to-be-confirmed) picture of L5 pyramidal neurons that are driven by basal instead of apical inputs. This is consistent with recent experiments that micro-dissected the dendritic tree, although of L2/3 neurons (Park, Papoutsi et al., bioRxiv, 2019). Somatic spikes would then propagate into the apical tree and, if elicited at high-enough frequency, recruit active dendritic sodium and calcium channels. The activation of the dendritic conductances could in addition depend on local apical input and play several roles, for example: 1) enhancement of the somatic spikes (bursting), 2) triggering of synaptic weight changes in the apical dendrite, 3) triggering of transmitter release to signal from apical locations to their respective pre-synapse. However, it would be unlikely that apical input by itself generates a calcium spike that activates the soma.

Conclusion: The studies by Francioni and Rochefort and Beaulieu-Laroche et al. contribute a lot to a better understanding of the function of apical dendrites of L5 pyramidal neurons. Even though the main result might seem like a partial disappointment at first glance, this kind of systematic study (instead of a study strongly driven by an hypothesis) seems to be the right thing to me. And more of the same, with different behavioral conditions, other cortical regions and possibly a more detailed analysis of the rare dendritic events seems to be the way to go. These two very interesting studies provide important information about somato-dendritic coupling, but the apical dendrite of L5 neurons still largely remains mysterious.

Posted in Calcium Imaging, electrophysiology, Microscopy, Reviews | Tagged , , , | 4 Comments

Post-publication review: “Precise excitation-inhibition balance controls gain and timing in the hippocampus”

It requires more than a quick look at the abstract and the figures to fully understand a research paper and its limitations. One way to go there is to write a summary or critical review of a paper. In a contribution to (informal) post-publication review, I will select neuroscience papers that are, in my opinion, worth the time needed to write up a review. On L5 apical dendrites [2]; on simulations of calcium imaging datasets [3].

Title: Bhatia et al., Precise excitation-inhibition balance controls gain and timing in the hippocampus, eLife (2019).

Target audience: Theoretical neuroscientists working on balanced networks and the role of inhibition; people working on the CA3 to CA1 feedforward circuit in hippocampus; people working on rules governing the connectivity of inhibitory interneurons; people interested in codes dependent on precise spike timing.

Why I chose this paper: I have worked on a similar topic at the interface of theoretical and experimental neuroscience, the so-called ‘precise balance of excitation and inhibition’ (Rupprecht and Friedrich, Neuron (2018)). Of course I’m interested what other experimental studies have found out about this important, although not very easily accessible concept.

graphical_abstract

From Bhatia et al., eLife (2019), used under CC BY 4.0 license (modified from Fig. 1).

Summary: The authors use a slice preparation to dissect the feedforward circuit from CA3 to CA1 in the mouse hippocampus (figure above; left part).

Two main findings.

The first finding (‘detailed balance’): They stimulate CA3 that expresses channelrhodopsin with an artificial set of patterned light stimuli (figure above; lower left and right parts). Then, they use whole-cell recordings to measure inhibitory and excitatory currents in single neurons in CA1. They find that even for very weak stimuli that presumably elicit few or even single action potentials only in CA3, an inhibitory current is evoked in CA1. Moreover, and even more interesting, the size of this inhibitory current matches the excitatory current not only on average, but also for individual stimuli. This feature has been termed ‘detailed balance’ by theoretical neuroscientists.

graphical_abstractii.png

From Bhatia et al., eLife (2019), used under CC BY 4.0 license (modified from Fig. 2).

The second main finding: The balanced inputs as recorded with voltage-clamp end up, however, in a non-linear integration as observed with current-clamp recordings. In particular, the authors observe a supra-linear relationship which they describe as ‘subthreshold divisive normalization’. In addition, they notice that inhibition kicks in faster in the CA3 neuron when the excitatory (and inhibitory) inputs to the CA3 neuron are stronger. They use this finding and simulations to support the idea that the reduction of inhibitory delays is the underlying cause of the observed divisive normalization.

In the following, I will only discuss the first finding. The second finding is already broadly discussed in the paper and also by the reviews (which are partly accessible, thanks to eLife!). In general, I find the first finding (detailed balance) much more interesting, whereas the main focus of the paper is on the second finding.

Consequences of a detailed balance: A detailed balance in this feedforward path from CA3 to CA1 is something unexpected, because it would require very specific feedforward connectivity of interneurons matching the feedforward connectivity of excitatory connections. Importantly, the authors put this in the context of previous work on CA1, which found that almost any CA1 cell can be converted into a place cell for any spatial location:

“Precisely balanced networks, with all input subsets balanced, are well suited for input gating (Barron et al., 2017; Hennequin et al., 2017). The finding that most silent CA1 cells can be converted to place cells for arbitrary locations predicts the existence of an input gating mechanism (Lee et al., 2012), but the nature of this mechanism remains unknown. One prediction of precise balance is that inputs for multiple potential place fields may be balanced, and hence place field activity is gated ‘off’. Evoked depolarizations (Lee et al., 2012) or dendritic plateau potentials (Bittner et al., 2015; Bittner et al., 2017), which potentiate the subset of active synapses, that is, change the I/E ratio (Grienberger et al., 2017), can flip the gate ‘on’, thereby converting a silent cell to a place cell for that specific place field”, write Bhatia et al., eLife (2019).

The devil’s advocate: Detailed balanced means, as mentioned before, that the size of an inhibitory current matches the excitatory current not only on average, but for individual stimuli. It is important, as I have described in my own experimental work (Rupprecht et al., Neuron (2018)), that this statement of a detailed balance can only be made if the different individual stimuli are equally strong.

It is therefore not a good idea to assess detailed balance by pooling over all stimuli with 1, 3, 5, 7 and 9 squares of optogenetic activation. Instead, one should compare within a set of stimuli with e.g. 1 square. This is done, as far as I understand it, only in Fig. 2h (example shown for a single neuron) and, based on a current-clamp dataset, in Fig. 2/Supplement 2e-f. It would have been interesting to analyze this aspect in more depth.

More importantly, even when analyzing e.g. only stimuli that all of them correspond to the stimulation of one “square” with photoactivation, this does not mean that all those stimuli are equally strong – which seems to be asssumed in the modeling part (Fig. 2/Supplement 2i,k). As I understand it, the most parsimonious explanation of the data would not be a detailed balance of excitatory and inhibitory inputs in this feedforward circuit, but a distribution of activation strengths for each stimulus set.

For example, stimulation of grid position 3 may overall result in 17 action potentials in CA3, whereas stimulation of grid position 7 may result in 55 action potentials and grid position 9 in zero action potentials. Feedforward inhibition could then simply integrate all action potentials in CA3 and send out a global signal proportional to this number. Indeed, even a single interneuron could, in theory, do this job. Therefore, it seems that the results can be explained by simple and plausible assumptions that do not require detailed balance. Altogether, I think that the paper provides some evidence for a detailed balance in the CA3 to CA1 feedforward circuit, but given the unexpectedness of this result, the evidence is a bit too weak to make me believe the conclusion entirely.

To fully test for a detailed balance, one could try to further analyze the existing data. However, a more direct way to test it would be much more convincing. This could be done by stimulation of CA3 and recording currents in CA1 as before, but at the same time recording the precise number of action potentials elicited in CA3 neurons. The best and maybe only reliable way to do so would be to perform paired recordings, with one (or more) neuron(s) patched in CA3 that are triggered to fire a given number of action potentials, while recording from one (or more) neuron(s) in CA1 in voltage-clamp. This would require considerably more work and result in much lower throughput than the opto-activation experiments, but would allow to address the question of a detailed balance directly and with higher precision.

Conclusion: Overall, I liked the study because it uses experiments to address an important question from theoretical neuroscience. The finding of a balance of excitation and inhibition in CA1 even for (presumably) very few spikes elicited in CA3 is an interesting and unexpected finding which is clearly supported by the data. I’m a bit less convinced about the evidence that this balance is also detailed. However, this does not affect anything of the remaining paper (which constitutes its main part). This main part investigates the supra-linear summation of inputs and the information contained in the timing of inputs (which I do not want to discuss here in any detail).

More context: What is detailed and precise balance? Hennequin et al., Annual Review of Neuroscience (2017). – A broadly accessible explanation of the supra-linear summation presented in the paper: NCBS News.

Posted in electrophysiology, Neuronal activity, Reviews | Tagged , , , | 2 Comments

Photon yield and pulse dispersion

This blog post serves as a link to a case report about the debugging of a two-photon microscope which showed a too low fluorescence yield. With the help of the internet, I singled down the cause of the problem to the dispersion of the femtosecond pulses by dielectric mirrors in the microscope. The lessons learned from that have been summarized before on Labrigger’s blog (also check out the comments!).

However, I think it makes sense to report this debugging attempt in a bit more detail, and probably there are two-photon microscope users who are keen on understanding more about the technical details and could profit from such a technical report. It includes not only nice pictures, but also several useful lessons that can be learned from that, some obvious or well-known, others not so much.

RandomMirrorOptimization

I used this occasion to try out a Github-based publication template provided by Andrew York. This template is written in a publication-like style, but in HTML, which allows to take advantage of some degree of interactivity and the embedding of web-links. The advantages compared to purely pdf-based publications are obvious; platforms like distill.pub show, although limited to the field of machine learning, how the future of publishing could look like: interactive, open, possibly based on web interfaces instead of a focus on print.

For me, using this template was an attempt to familiarize myself with this sort of publishing, and to test how difficult is to use such a template. I hope you enjoy the read:

Pulse dispersion, dielectric mirrors and fluorescence yield: A case report of a two-photon microscope.

Posted in Calcium Imaging, Imaging, Microscopy | Tagged , , , | 3 Comments

The power of correlation functions

During my physics studies, I got to know several mathematical tools that turned out to be extremely useful to describe the world and to analyze data, for example vector calculus, fourier analysis or differential equations. Another tool that I find particularly useful for my current work as a neuroscientist and which is, however, rarely mentioned explicitly are correlation functions. In the following, I will try to give an intuition of the power of correlation functions using a couple of examples.

.What are correlation functions?

To put it in very simple terms, a correlation coefficient measures how similar two signals (A and B) are after being normalized. Different from correlation coefficients, correlation functions are not single values, but functions of two input signals A and B. This can be a correlation function C_{AB} of a time lag, C_{AB}(\tau), or of a distance in space, C_{AB}(\Delta x) . The correlation function at a time lag or distance of zero, recovers the correlation coefficient, C_{AB}(0), except for a normalizing factor.

The value of a correlation function at a given value of \tau or \Delta x indicates how similar the two input signals A and B are when one of the signals is shifted in time by \tau or in space by \Delta x.

To make the result of this operation more clear, here are two simple examples, with signals A(t) (black) and B(t) (gray) as noisy sine waves that are in phase (left) or out of phase (right):

Principle

While the cross-correlation function peaks at a time lag of \tau = 0 for the synchronous case, the peak is shifted to \tau \neq 0 for the out-of-phase case. The value at a time lag of 0 is proportional to the correlation coefficient: a high value for the left side, a value close to zero for the right hand side. Also note that the correlation function used averaging over the full signal duration to get rid of the noise.

Computing the correlation function C_{AB} in Matlab or Python

Computing the correlation function is actually straightforward in Matlab or Python. Pay attention to subtract the mean of the two signals (e.g., B = B – mean(B) in Matlab or B = B – np.mean(B) in Python), because the functions expect normalized input.

Matlab:

A = rand(1000,1);
B = rand(1000,1);
C = xcorr(A,B,'unbiased');

Python:

import numpy as np
A = np.random.norm(0,1,1000)
B = np.random.norm(0,1,1000)
C = np.correlate(A,B,'full')

or

import scipy.signal as signal
C = signal.correlate(A,B)

It is a good but a bit tedious exercise to write one’s own cross correlation function in a basic programming language. Usually the normalization at some point can cause headaches.

1. Spatial correlation functions for image registration

In microscopy, there’s often the problem to map two images onto each other. The following examples are two average images of the same brain region, recorded at different time points and therefore shifted meanwhile due to drift. I included a horizontal line for orientation:

Image_registratoin.png

To find out the drift, we can use correlation functions, measuring the similarity of the two images for all possible shifts, with the result that the shift in x-direction is 0 pixels, whereas the shift in y-direction is 4 pixels (here in Matlab):

movie_AVG1; % average image 1
movie_AVG2; % average image 2
result_conv = fftshift(real(ifft2(conj(fft2(movie_AVG1)).*fft2(movie_AVG2))));
[y,x] = find(result_conv == max(result_conv(:)));
shift_y = y - ( size(movie_AVG1,1)/2 + 1 )
shift_x = x - ( size(movie_AVG2,2)/2 + 1 )

Here, I calculated the correlation function using fast fourier transforms, taking advantage of a simple mathematical property of correlation functions. I could also have done the same computation with the built-in function xcorr2(movie_AVG1,movieAVG2) in Matlab, which is however much slower and requires subtraction of the respective mean from the images.

Similar algorithms are used for most image registration functions in ImageJ, Python or Matlab.

2. Local spatial correlation functions for particle image velocimetry

To go one step further, one can also compute a local instead of a global shift, for example if there are any deformations of the images that result in local deformation fields.

A more interesting application of the same principle of local deformations is a method referred to as particle image velocimetry (PIV), which has been developed in the field of experimental fluid mechanics. Using a sequence of images, correlation functions are used to extract local flow fields, as well as sinks and sources of the observed transport phenomenon. Here is, from work for my Diploma thesis, an example movie of a one-cell C. elegans embryo just before the first cell division, observed using DIC microscopy. I used the granular stuff in the cytoplasm to track the cytosolic flow patterns using PIV (with the toolbox PIVlab). The overlaid yellow arrows indicate the (wildly changing) direction of the local cytosolic flow field:

PIV_1.gif

3. Temporal cross-correlation functions

One of most fascinating usages of cross-correlation functions for analysis of experimental data is for fluorescence cross-correlation spectroscopy (FCCS), or its more commonly used simpler version, fluorescence correlation spectroscopy (FCS), the latter of which is based on auto-correlation instead of cross-correlation functions.

Peri-stimulus time histograms (PSTHs) are a much more basic analysis tool that is commonly used by electrophysiologists to quantify the occurrence of a quantity triggered by certain events. Sometimes, events as ill-defined as the crests of an oscillatory signal are used as a trigger for a PSTH. Using correlation functions gets rid of this mess by measuring how much a quantity is affected depending on the quantitative history of the trigger signal.

In electrophysiologal work published in 2018, I used correlation functions to measure the phase relationship between an oscillatory local field potential (LFP) signal and an oscillatory component in a simultaneous whole-cell recording (for details, check out a part of figure 7 in the paper):

Temporal_correlation

4. Autocorrelation functions for time series analysis

Auto-correlation functions are not only a tool for non-intuitive experimental methods like FCS, but also perfect to quantify periodicities in a time series. For example, if there is an oscillatory behavior in a swim pattern of a fish, in the firing of a neuron or in the spatial density of clouds, autocorrelations can easily quantify this periodicity.

Here is an example, again from an LFP recording. On the left, the signal seems clearly oscillatory, but how can we properly quantify the oscillatory period? We use an auto-correlation function, and the peak at around 40 ms in the plot on the right clearly indicates the oscillatory period (black arrow):

Autocorrelation

Correlation functions in physics

If you find the above examples interesting and want to understand what correlation functions can be used for, it could be a good idea to dive into physics, where correlation functions are all over the place:

In addition, the mathematical aspects of correlation functions are quite rewarding to explore, for example the intimate relationship between auto-correlation functions and power spectra.

As another interesting use of auto-correlation functions, the fluctuation-dissipation theorem gives an idea how spontaneous fluctuations of a system close to thermodynamical equilibrium can predict the linear response of the system towards external perturbations. It’s a bit discouraging for biologists to understand that this theorem can hardly be applied to biological systems, which live far from thermodynamic equilibrium and which show responses that are rarely linear.

Still, it is amazing to see what physics can do with correlation functions and how powerful correlation functions are at extracting precise measurements from sometimes very noisy data.

Posted in Calcium Imaging, Data analysis, electrophysiology | Tagged , , , , | 4 Comments

Annual report of my intuition about the brain

There are not many incentives for young neuroscientists to think aloud about big questions. Due to lack both of knowledge and authority, discussing very broad questions like how the brain works risks to be embarrassing at best. Still, I feel that not doing so is even more detrimental since it restrains the potential for internal development: Exposing one’s thoughts comes with the potential to refine them and to dissociate from them, thereby bringing down or advancing ideas that might have got stuck.
I want to make it a habit to report some of the thoughts about the brain that marked me most during the past twelve month at the end of each year, with the hope to advance and structure the progress in the part of my understanding of the brain which is not immediately reflected in journal publications. Other year-end write-ups: 2018, 2019, 2020, 2021.

How I got interested in dendrites

The lines of thought described in the following actually go back as far as to 2015. I was planning to switch from calcium imaging to whole-cell recordings as my main laboratory technique and started understanding the power of studies relying on this technique. In summer 2015, I came across a paper by Katie Bittner in Jeff Magee’s lab [1] (followed up by another paper [2]). Those papers showed that electrical “plateau potentials” can drive the formation of a place cell within a single trial. The authors established in vivo whole-cell recordings deep in the CA1 region of the hippocampus. Using this technique, they could generate plateau potentials by somatic current injection and thereby trigger the generation of a place field. As probably many others, I was immediately struck by this single-shot learning behavior, but, also due to lack of background knowledge, I was not yet able to see it in a larger context.

Later, when I was searching for potential postdoc positions in 2018, I first fully encountered the mystery of the apical dendrites of pyramidal neurons. Pyramidal neurons in layer 5 of the mammalian cortex grow from their soma a “basal” dendritic tree that remains rather local in layer 5, and in addition a thick “apical” trunk that goes up to layer 1, where it branches into many small apical dendritic processes (the apical “tuft”).

I was particularly intrigued by a review by Matthew Larkum from 2013 suggesting a specific function for the apical tuft of L5 neurons. This suggested function would be to detect almost coincident somatic activity and strong input to apical dendrites, resulting in a calcium spike in the apical trunk and leading to somatic bursting [3].

Problem 1: Top-down input to the apical dendrites of pyramidal neurons

Apical dendrites of L5 neurons are thought to receive mainly top-down input, whereas the basal dendrites are predominantly contacted by bottom-up input. For example, basal dendrites of the primary visual cortex would be expected to receive more sensory input from the primary thalamic region, whereas apical dendritic processes would receive rather context-related input from brain regions higher in the sensory hierarchy. I do not know how well this separation of top-down and bottom-up inputs for apical and basal dendrites holds true – in an earlier blog post I have described why I am generally not a fan of hierarchies like this top-down/bottom-up connectivity scheme, although I still find it a fascinating idea.

Since I’m currently working next door to the lab of Georg Keller, who is interested in predictive processing in visual cortex (check out his 2018 review [4]), I could not prevent myself from wondering whether this top-down contextual input to the apical dendrites could simply be predictions. This possibility is also mentioned in the Larkum review [3]. However, in the theory of predictive processing, predictions (here: apical input) should be subtracted from the sensory input (here: basal input), or the sensory input should be subtracted from the predictions. As mentioned above in the review by Matthew Larkum, however, the apical trunk seems to compute the coincidence of those inputs rather than the difference. Therefore, this somehow does not seem to make sense.

There are ideas how to implement predictive processing using L5 pyramidal cells nevertheless. For example, there is an interesting computational model that is pretty detailed (described by Sacramento et al. [5]). The idea here is that the apical compartment does not simply signal top-down input, but encodes an error signal between local inhibitory signals and top-down excitatory input. Some assumptions of this model seem to be unrealistic and many aspects of the model are simply unconstrained by experiments, but it is an interesting starting point nethertheless.

Problem 2: Coupling between apical dendrites and the soma

Overall, this leaves me with the impression that the apical compartment might be something crucial to understand. The separation of processing in apical and somatic compartments is an assumption that seems legit given the large electrotonic distance between soma and the apical dendrite. In addition, this assumption is supported by experimental data (e.g., Cichon & Gan, 2015 [6]; Seibt et al., 2017 [7]; and some other studies). However, for all of those studies, no direct evidence for the decoupling of somatic and apical activity was available. Direct evidence would mean simultaneous recording of somatic and dendritic activity, which is challenging even for an indirect method as calcium imaging due to the large spatial separation of soma and distal apical dendrite. Probing of calcium dynamics in a direct way so far has not shown strong decoupling of somatic and dendritic activity (e.g., Helmchen et al., 1997 [8]; or more recently Kerlin et al., 2018 [9]). To be more precise, these studies showed that almost all calcium events in the apical dendrites – with very few exceptions – were temporally coupled to backpropagating action potentials. This seems to be somehow at odds with the idea of separate processing in somatic and apical compartments.

Of course, this is only about dendritic calcium signals, not about the voltage. Recording of the voltages over multiple locations of a dendritic tree, for which there is currently no reliable method, could potentially result in a different picture. Plus, the brain areas and behavioral contexts are not immediately comparable between the behavioral tasks in these experiments. For example, Helmchen et al. [8] used anesthetized rats; Kerlin et al. [9] trained their mice extensively before experiments; Cichon et al. [6] recorded dendritic activity during a weird learning paradigm that might have resulted in a lot of confusion in the mice; and Seibt et al. [7] focused on dendritic activity in mice and rats during sleep.

As a result of these (seemingly) contradictory results, I’m intrigued by the  unresolved question how tightly the activities of soma and apical dendrites of L5 neurons are indeed coupled. Or rather, under which circumstances both compartments become uncoupled. The answer to this question is completely unclear to me.

Problem 3: What do bursts of pyramidal neurons signal?

It is however clear that somatic action potentials to some extent invade the apical dendritic tree. This does not seem to be a random side effect, since it was reinforced by evolution by the insertion of active conductances into the dendritic tree. One possible purpose of this backpropagating action potential could be to activate the inputs of the apical dendrite, resulting in non-linear amplification in the distal dendrites or in the apical trunk (as described by the Larkum review [3]) and thereafter in somatic burst firing. What is the purpose of these bursts? I can come up with two possible explanations:

(1) As suggested by the experiments in the Magee lab ([1][2]), the bursts could be a strong intracellular signal to reinforce recently activated context synapses. – If so, in which synapses would plasticity occur, in basal or rather apical dendrites? In the studies from the Magee lab in the hippocampus plasticity in synapses of the stratum radiatum of CA1 was observed [2]; those synapses are thought to provide spatial context. How would this translate to cortex?

(2) A second possible function of bursts could be to signal not inside of a neuron, but between neurons. Regular spiking is ideally suited to drive postsynaptic neurons with depressing synapses, i.e., only the first spike of a rapid sequence of spikes would trigger substantial synaptic release of neurotransmitters. Bursting, however, is ideally suited to drive postsynaptic neurons that are connected via facilitating synapses. The bursts would therefore be a very sparse code that could signal a coincidence of somatic spiking and apical input to the downstream neuron. In a theoretical study, Naud & Sprekeler [10] investigated the potential of such multiplexing through simple spikes and bursts for separate processing of top-down and bottom-up input in a hierarchical network. And Blake Richards mentioned (in a talk that I’ve watched on youtube, start at min 22:14), while not going into the details, the possibility to use this multiplexing for helping to solve the “credit assignment problem”.

Brief digression: The credit assignment problem is about the question how a neuron somewhere in the brain network can learn to weigh the incoming information in order to become better at a given task. This problem is also addressed by the previously mentioned paper by Sacramento et al. [5], and there is a paper by Guerguiev et al. [11] that goes into a similar direction but is a bit chaotic. Lillicrap and Richards just published a review on how the credit assignment could be solved using (apical) dendrites [12]. It is important to note that both Sacramento et al. and Guerguiev et al. suggest solutions that are approximations of “backpropagation” of errors, the solution to the credit assignment problem that has been fameously found for artificial neural networks. Backpropagation, however, follows a linearized gradient in error space and therefore only allows for small synaptic changes – and thus does not really allow for the single-shot learning behavior that has been observed in animals ([1][2]). Therefore I’m not sure whether it is good idea to search for an implementation of an algorithm similar to backpropagation in the brain.

Problem 4: It’s getting ever more complex

In addition to the open questions mentioned above, some other points related to the function of apical dendrites are also not clear.

For example, what role do inhibitory neurons play that specifically target the apical dendrite? With a disinhibtory circuit motif, interneurons could specifically gate plasticity by blocking inhibition of an apical dendrite (check this review by Letzkus et al. [13]). Following this line of thought, it is, however, not clear to me whether disinhibition is (branch- or neuron-) specific or rather a broad, global gating mechanism of plasticity that allows for specific plasticity by other means.

As another example, it is to some extent clear how the membrane potential behaves in vivo in the soma – but less so in the dendrites of the very same neurons. Dendrites might integrate much fewer inputs than a soma and thereby exhibit much stronger voltage fluctuations – unless there is a precise local balance of excitatory and inhibitory inputs to a single dendrite (this question is based on work I did in zebrafish). A recent study addressed this question of balancedness partially by mapping the co-localization of excitatory and inhibitory neurons on a full tree of L2/3 pyramidal neurons [14].

In the context of balanced networks, I’m also wondering whether apical dendrites in living, unanesthetized brains operate in a high-conductance state as a result of strong excitatory and inhibitory inputs, which has been suggested for balanced networks. If so, I would be interested to know how the mass of open input channels in vivo would affect the coupling between dendritic segments compared to ex vivo slice studies. For this particular question, I’m not sure whether I’m just ignoring existing literature on the subject or whether these questions have simply not yet been addressed experimentally.

Summary

What happens in apical dendrites of L5 pyramidal neurons remains mysterious to me, in particular if the neuron is integrated in the cortex of a behaving animal or human being. I do not know which kind of events trigger activity of the apical dendrite, and I do not know under which circumstances and how the apical compartment communicates with the soma. I wonder how well the compartments are connected electrically in vivo. And it is an open question how both synaptic plasticity and non-plastic information processing are affected by activation of the apical trunk or the more distal apical tuft. Despite this lack of knowledge, I would guess that understanding apical dendrites is not sufficient, but probably necessary to understand what a cortical region does as a whole.

Maybe I’m wrong about some of my interpretations; probably I’m overlooking some import studies. If something is wrong in my guesses and interpretations, or if I am missing an important piece of experimental or theoretical evidence, please let me know. I do not have an agenda that I want to defend but instead would like to understand. Therefore, critical comments are even more welcome than positive feedback!

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References

[1] Bittner KC, Grienberger C, Vaidya SP, Milstein AD, Macklin JJ, Suh J, Tonegawa S & Magee JC. Conjunctive input processing drives feature selectivity in hippocampal CA1 neurons. Nature Neuroscience (2015).

[2] Bittner KC, Milstein AD, Grienberger C, Romani S, & Magee JC. Behavioral time scale synaptic plasticity underlies CA1 place fields. Science (2017).

[3] Larkum ME. A cellular mechanism for cortical associations: an organizing principle for the cerebral cortex. Trends in Neurosciences (2013).

[4] Keller GB, & Mrsic-Flogel TD. Predictive Processing: A Canonical Cortical Computation. Neuron (2018).

[5] Sacramento J, Costa RP, Bengio Y, & Senn W. Dendritic cortical microcircuits approximate the backpropagation algorithm. Advances in Neural Information Processing Systems (2018).

[6] Cichon J, & Gan WB. Branch-specific dendritic Ca2+ spikes cause persistent synaptic plasticity. Nature (2015).

[7] Seibt J, Richard CJ, Sigl-Glöckner J, Takahashi N, Kaplan DI, Doron G, Limoges D, Bocklisch C, & Larkum ME. Cortical dendritic activity correlates with spindle-rich oscillations during sleep in rodents. Nature Communications (2017).

[8] Helmchen F, Svoboda K, Denk W, & Tank DW. In vivo dendritic calcium dynamics in deep-layer cortical pyramidal neurons. Nature Neuroscience (1999).

[9] Kerlin A, Mohar B, Flickinger D, MacLennan BJ, Davis C, Spruston N & Svoboda K. Functional clustering of dendritic activity during decision-making. bioRxiv (2018).

[10] Naud R, & Sprekeler H. Sparse bursts optimize information transmission in a multiplexed neural code. PNAS (2018).

[11] Guerguiev J, Lillicrap TP, & Richards, BA. Towards deep learning with segregated dendrites. eLife (2017).

[12] Richards BA, & Lillicrap TP. Dendritic solutions to the credit assignment problem. Current Opinion in Neurobiology (2019).

[13] Letzkus JJ, Wolff SB, & Lüthi A. Disinhibition, a circuit mechanism for associative learning and memory. Neuron (2015).

[14] Iascone DM, Li Y, Sumbul U, Doron M, Chen H, Andreu V, Goudy F, Segev I, Peng H, & Polleux F. Whole-neuron synaptic mapping reveals local balance between excitatory and inhibitory synapse organization. bioRxiv (2018).

Posted in Calcium Imaging, electrophysiology, machine learning, Neuronal activity | Tagged , , , , | 4 Comments

Whole-cell patch clamp, part 4: look and feel

In previous blog posts, I have been discussing some aspects of whole-cell patch clamp recordings ([1], [2], [3][4]). Today, I will show some instructive videos that I recorded during experiments. I’m hoping that they will convey the look and feel of the procedure of whole-cell patching in an intact brain using two-photon microscopy to target neurons.

Two-photon targeted patching is the core method of my recent paper on a precise synaptic balance of input currents, focusing on biological questions instead of methods (Rupprecht et al. (2018)). The underlying method has been described before as “shadow-patching”. Dye ejected from the pipette allows to visualize cell bodies as dark shadows in a sea of fluorescence (see Kitamura et al. (2008); also check out Judkewitz et al. (2009)Margrie et al. (2003) and Komai et al. (2008)). Although these papers are very useful resources, they do not allow to understand how the procedure of patching a neuron looks and feels like to the experimenter.
For camera-based whole-cell patch clamp recordings in slices or dissociated cultures, on the other hand, there are a handful of videos on the internet (for example this one). However, this looks quite different from patching targeted by two-photon imaging. Camera-based imaging only allows to patch in thin tissues like cultures or slices since camera-imaging does not provide good optical sectioning and penetration. Here, I will show (uncut) movies of the process of patching, while monitoring the applied pressure and the test pulses.

shadow_patchx-01x1-e1540652954642.png

As a brief introduction to two-photon targeted shadow-patching, the pipette approaches the brain surface while blowing out dye (1). After entering the tissue and after lowering of the pressure (2), the pipette closely approaches a target neuron (3). After gigaseal formation and break-in, allowing for electrophysiological recordings, the targeted neuron fills with the dye while the surroundings turn dark again (4).

All of the videos below are patch clamp recordings in the olfactory cortex homolog of zebrafish in an ex vivo preparation where the entire brain including the nose remain intact. Neurons are labeled using pan-neuronal expression of GCaMP6f, which is barely visible compared to the dye; shadow-imaging is performed using a resonant scanning microscope (described before on this blog). Neuronal somata in this brain area are quite small (5-8 μm in diameter, which is probably half or less the size of a typical mouse principal neuron), which can render targeted patching quite difficult, especially in deeper structures, where resolution degrades due to scattering. All of the recordings below are in more or less superficial regions (<200 μm below the brain surface). Patching deeper neurons usually required much more focused attention from my side, and the pipette tip could not be localized as easily as in the movies below.

For the paper, I produced a “Methods Video”, which due to restrictions from Neuron is limited to a duration of 1 min. I wanted to record not only the fluorescence movie during patching, but also the pressure and the test pulses applied to the electrode. For screen capture, I used the software TinyTake; for video editing, KdenLive (Linux); for text to speech synthesis of the next video, Wavenet provided by Google Cloud, which I have discussed before on this blog). The video is available in the Star Methods section of the paper, and also here:

However, the short duration of the video is maybe appropriate as a short visual summary for a paper, but not ideal for somebody who wants to get an intuition on how shadow patching can be done in reality. Therefore, here’s a longer excerpt of the same recording. I sometimes use this excerpt for presentations:

Still, this is a bit too condensed. Therefore, below you will find the uncut version of this particular patching experience. I admit it is really boring to watch, but I think it is also instructive. Not shown in the video are the changing positions of the two micromanipulators that move the pipette tip and the focus of the microscope; also not shown are small modifications to the laser power, zoom settings, bidirectional scan phase or the electrophysiological recording conditions. And yes, I’m aware that this recording is far from being perfect, but I think it can still be a useful starting point for a prospective electrophysiologist.

Next comes the patching of a different neuron. Usually, I’m using a syringe to apply pressure and suction to the pipette (other people prefer to apply the suction with the mouth). Here, after establishment of the giga-seal, the syringe somehow broke down and was not useable any more. I quickly constructed a temporary mouthpiece out of some tubings and finally managed to successfully break into the cell.

And here yet another successful attempt:

In total, I made around 20 such simultaneous recordings of the two-photon video, the pressure indicator and the test pulses window. Assembling the videos, however, turned out to take quite some time, and therefore I will show only one more movie, this time of a failed attempt. Almost immediately after entering the tissue, I realized that this recording would probably be not successful (the dura covering the brain sticked to the pipette tip for too long). Usually I would have stopped the attempt as early as possible in order not to waste time. In this case, I still tried to patch a neuron in order to get a nice recording of a failed attempt. Failures are not really rare when you try patching, especially in deep and small neurons.

Of course, shadow-patching might look somewhat different in a different brain region or with a different microscope or at a different tissue depth. To give you an idea, here is a recording with lower light levels due to lower dye concentration and laser power and with some problems related to the microscope (which I was too lazy to debug back then) which did not allow to zoom in as much as for the previous recordings. For someone not familiar with the particular setup, it is probably quite difficult to accurately see the pipette tip – which is crucial to move the pipette to the right location, in particular in the z-direction.

Posted in Calcium Imaging, electrophysiology, Imaging, Microscopy, Neuronal activity, zebrafish | Tagged , , , | 3 Comments

Precise synaptic balance of excitation and inhibition

The main paper of my PhD just got published: Rupprecht and Friedrich, Precise Synaptic Balance in the Zebrafish Homolog of Olfactory Cortex, Neuron (2018). (PDF)

You might like it if you are also interested in

  • Classical balanced networks
  • Things you can do with whole-cell voltage clamp
  • Olfactory cortex
  • More recent ideas about balanced networks
  • Coordination of excitatory and inhibitory synaptic inputs in single neurons
  • Adult zebrafish

To summarize this work in one sentence, this is a study of the coordination of excitatory and inhibitory synaptic inputs in single neurons. If you want to know the details, you should definitely read the paper.

The main part of the study is purely experimental, but one of its strengths is that it connects the experimental findings with computational concepts about balanced networks. The concept of a balanced state has been brought up in the mid-90s by Shadlen & Newsome and van Vreeswijk & Sompolinsky (among others). More recent theoretical work has, in my opinion, contributed a lot to identifying and correcting some weaknesses of the classical balanced network, and has come up with new concepts about circuit function of balanced networks that are of general interest to those who want to understand how the brain works. If you’re interested in a discussion of these concepts, I can recommend the following review articles as starting points (which are also discussed in our paper):

  1. Denève S, Machens CK, Efficient Codes and Balanced Networks, Nature Neuroscience (2016).
    .
  2. Hennequin G, Agnes EJ, Vogels TP, Inhibitory Plasticity: Balance, Control, and Codependence, Annual Review of Neuroscience (2017)..
    .

But let’s for one moment think beyond the scope of this work, which focused on synaptic inputs on the single-cell level – let’s think about the subcellular level. One thing I’d be interested in would be to have a closer look at the coordination of synaptic inputs on small dendritic segments instead of entire neurons. There is already a handful of studies that go into that direction, using mechanisms of synaptic plasticity (Chiu et al., Neuron, 2018) or the anatomical distribution of synapses (Iascone et al., bioRxiv) as entry points.

I’m really looking forward to seeing more research going into this subcellular level of neuronal processing. I can understand that people find population codes as observed by calcium imaging and extracellular recordings of interest, especially with respect to behavior. But I’m also convinced that mechanistic insights into how neurons work can be better obtained by investigating cellular and sub-cellular processes. Our published study investigates a variety of details on the cellular level; but this is only a small fraction of the many things that go unnoticed if you only look at the firing of neurons and not at underlying processes, for example the synaptic inputs.

Posted in Calcium Imaging, Data analysis, electrophysiology, Network analysis, Neuronal activity, zebrafish | Tagged , , , , | 1 Comment

Alvarez lenses and other strangely shaped optical elements

In typical microscopes, lenses or mirrors are moved forth and back to change the position of their focus. Tunable lenses like the electro-tunable lens or the TAG lens, on the other hand, are deformed by an external force and thereby change their focal length. One interesting concept that I had not noticed until recently is the idea of the Alvarez lens, named after its inventor (described in this 1964 patent). I came across it in a 2017 paper from the lab of Monika Ritsch-Marte from Innsbruck/Austria. The following picture adapted from their paper very nicely illustrates the effect:

drawing

By lateral displacement of the two lens elements against each other one can focus or de-focus the beam. In two papers from this lab (paper 1, paper 2), the authors used a method that sort of replaces this lateral (slow) movement with a (fast) rotation of a galvo mirror by using a creative optical configuration (check out the paper for the details, it is a pleasure to read).

There a couple of things to notice: The Alvarez lens is a bit more complex in 2D (the above schematic illustrates a (de-)focusing system for 1D only). The authors use diffractive instead of refractive Alvarez lenses. They use only visible light (no near-infrared light, which I would prefer). And they mention some other shortcomings of their approach.

Still, I find the principle very interesting and inspiring, and I hope that somebody will invest his or her time to put a system together that is not only a proof-of-principle, but an optimized system that reaches the best possible performance. This would probably also be a nice playground for a study of optical modeling and optimization: to find out which shape of the lens could perform much better than the Alvarez lens (like this study, but a bit more systematic with respect to possible lens surfaces).

Overall, this is a fascinating piece of optics, and I got interested also because I had always been intrigued by optical scanning methods where a simple movement of the beam is translated to a complex scanning scheme by an optical element (see for example this blog post on entirely passive scanning at MHz rates). For a long time, I hoped that a method similar to an Alvarez lens and based on a strangely shaped mirror (or lens) surface could be used to transform a linearly scanned pattern into something more complex (like a spiral scan, or a 2D raster scan). In theory, this is possible, but in practice the finite beam diameter would create a lot of problems. In addition, constructing an arbitrarily shaped mirror with good surface flatness and broadband reflective coatings would be quite costly.

One field where I long thought that such an approach could be applied, because it would be applicable for many microscopes, is the un-distortion of the non-linear angular scanning trajectory of resonant scanners (described in detail in a previous blog post). The idea would be that an optical element (the ‘black box lens’ in the schematic below) placed after the resonant scanner would somehow convert the angular dependency \sim \sin(\omega t) into a relationship that is rather linear in time, \sim t. Such that at time points close to the turnaround of the sine (blue time point below), the ‘black box lens’ would increase the angular deflection angle, eventually inversing the sine function:

Scan_impossible

I have the suspicion that this problem is practically not solvable due to the finite beam diameter, but it would be interesting to know whether there is a solution for this problem at least for the assumption of infinitely small scanning beams using geometric optics. This could be done by a lens whose diffractive power increases with distance x from the center of the lens.

Let’s assume a scan angle \alpha = \sin( \omega t). The scanned beam hits the black box lens at a position x(t) = \tan(\alpha) \cdot d with the distance d between the resonant scanner and the lens. The refractive power f(x) of the lens must therefore change with x such that the outgoing beam is linear in time. In approximative ABCD optics:

\left( \begin{array}{cc} 1 & 0 \\ -\frac{1}{f(x(t))} & 1 \end{array} \right) \cdot \left( \begin{array}{cc} x(t) \\ \alpha (t) \end{array} \right) \stackrel{!}{=} \left( \begin{array}{cc} x(t) \\ t \end{array} \right)

This results in the following expression for the local radius of the lens depending on the location x:

f(x) = \frac{x}{\arctan(x/d) - \arcsin(\arctan(x/d))}

You can find the equations and some plots also in a Jupyter notebook on Github. For small absolute values of x, f(x) diverges, indicating an infinite curvature, i.e., a lens that simply transmits light without deflection. With increasing/decreasing x, f(x) tends to zero, indicating increasing refractory power and a stronger local curvature of the lens surface.

Such a lens would only work optimally at one single zoom setting, which is probably one of the many reasons why nobody ever has tried this out. But it’s still interesting to think about it.

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Entanglement of temporal and spatial scales in the brain, but not in the mind

In physics, many problems can be solved by a separation of scales and thereby become tractable. For example, let’s have a look at surface waves on water: they are rather easy to understand when the water wave-length is much larger or much smaller than the depth of the water, but not if both scales are similar (wikipedia).

To give another example, light scattered by small particles (like fat bubbles in milk, or water drops in a cloud) can be described more easily if the wavelength of the light is much larger (Rayleigh scattering) or much smaller than the particles, but not if it is of the same order of magnitude (Mie scattering). Separation of scales is often key to making a problem tractable by mathematics.

What physicists like even more than separation of spatial scales, is the separation of different temporal scales. For example, consider two variables A(t) and B(t) that are influenced by each other:

\tau_1 \frac{d A}{d t} = f(A,B) \\ \\ \tau_2 \frac{d B}{d t} = g(A,B)

If the timescales separate, for example \tau_1 \gg \tau_2, the variable B(t) is basically seen as constant by the variable A(t). In this case, the variables can be decoupled, and the problem is often solvable. (Sidenote: In very simple and idealized systems without separations of scales, for example during some sort of phase transitions, mathematical physics can still come to the rescue and provide some clean solutions. But in most systems, this is not the case.)

I am convinced that problems do not become easier by a separation of scales only for physics or mathematics. I think that this applies even more to our intuition and our own understanding of the world. Automatically, we try to disentangle systems by using hierarchies and separations of length- and timescales, and if we are unable to do so, our intuition fails, as does the physics analysis.

What about the brain? In my opinion, the brain is one of those system that will defy human attempts to understand it by separating temporal processes or spatial modules. The brain consists of an enormous amount of different temporal and spatial scales that, however, overlap with each other and cannot be easily segregated. For example, on the timescale of few 100 ms, many different processes are non-stationary and therefore relevant at the same time: neuromodulation of many kinds; spike frequency adaptation and  presynaptic adaptation and facilitation; diffusion of proteins across spines, or ions across dendrites; calcium spikes; NMDA currents; et cetera. At a timescale of 1000 ms or 10 ms, it is a different but overlapping set of processes that are non-stationary. To put it short, it seems likely to me that the brain consists of a temporal and spatial continuum of processes, rather than a hierarchy.

Why would this be so? Because, as far as I can see, there is no incentive for nature to prevent the entanglement of temporal and spatial scales of all those processes. In contrast, those interactions may offer advantages that emerge randomly by evolution, at the cost of a higher complexity. Nature, which does not need to understand itself, probably does not care much about an increase of complexity, unlike the biologists working to disentangle the chaos.
It is perhaps misleading to personify ‘nature’ and to speak of an ‘incentive’. It is probably more acceptable to derive these processes from ‘entropic forces’, which make any ordered system, including the organic and cellular systems invented by evolution, less ordered and therefore more chaotic over time. Even if there was order once (think of a glass of water which is strictly colored green in the left and blue in the right half), random changes, which is the driving force of evolution, will undo this order (nothing can prevent that green and blue water will mix over time by random motion of its molecules, that is, diffusion).

In addition to the deficiency of our mind and of mathematical tools when it comes to entangled scales, I suspect based on personal experience that humans are to some extent unable to bring together knowledge from different hierarchies. In neuroscience, most researchers stick to one small level of observation and the related processes; and in most cases it is very difficult to bridge the gaps between levels. For example, “autism” can be addressed by a neurologist who thinks about case studies and very specific behavioral observations of her patients; by a geneticists looking for combinations of genes that make a certain autistic feature in humans more likely; or by a neurophysiologist studying neurons in animals or in vitro models of autism, trying to dissect the contribution of neuronal connectivity or ion channel expression. Many people believe (or hope) that with sufficient knowledge and understanding, these different levels of observation will fuse together, resulting in a complete understanding that pervades all levels. I would argue – and I’d like to be disproven – that a more pessimistic view seems to be more realistic and that humans will probably never achieve an understanding of neuronal circuits and the brain that is deep enough to bridge the gaps between the levels.

The limitations of both our mathematical tools and our mind when it comes to complex systems is obvious when we think of deep learning. For this field of machine learning, other than for the brain, we know all the basic principles (because we have defined them ourselves): Back-propagation of errors, gradient descent algorithms for optimization, weight-sharing in convolutional networks, rectified linear units (or maybe LSTM units), and a few more. Compared with the brain, the system is not very complex, and we can observe everything throughout the process without interfering with its operation. Still, although the process is 100% transparent, people struggle and fail to understand what is happening and why. There does not seem to be a simple answer to the question how it works. “What I cannot create, I do not understand”, Feynman famously wrote. But the act of creation does not automatically come with understanding.

Experimental neuroscience might face similar, but probably even more complex problems. The way to “understand” a neuronal process that is accepted by most researchers is a (mathematical or non-mathematical) model that can both reproduce and predict experimental results. However, if biology indeed consists of many processes and components that are entangled in space and time, also a model needs to be built that is entangled on several temporal and spatial scales. This can be done – no problem. However, this model will again resist attempts by mathematics or human intuition to understand it, similar to our current lack of understanding of the less complex deep networks. Therefore, the machine (the model, the computer program) will still be able to deal with the complexity and “understand” the brain, but I am not sure that human intuition will be able to follow.

I don’t want to deny all the pieces of progress that have been made to achieve a better understanding of the brain. I rather want to point out the limitation of the human mind when it comes to putting the pieces together.

Posted in Data analysis, machine learning, Network analysis | Tagged , , , , | 2 Comments