Abstract

Neurons are the computational elements that compose the brain and their fundamental principles of activity are known for decades. According to the long-lasting computational scheme, each neuron sums the incoming electrical signals via its dendrites and when the membrane potential reaches a certain threshold the neuron typically generates a spike to its axon. Here we present three types of experiments, using neuronal cultures, indicating that each neuron functions as a collection of independent threshold units. The neuron is anisotropically activated following the origin of the arriving signals to the membrane, via its dendritic trees. The first type of experiments demonstrates that a single neuron’s spike waveform typically varies as a function of the stimulation location. The second type reveals that spatial summation is absent for extracellular stimulations from different directions. The third type indicates that spatial summation and subtraction are not achieved when combining intra- and extra- cellular stimulations, as well as for nonlocal time interference, where the precise timings of the stimulations are irrelevant. Results call to re-examine neuronal functionalities beyond the traditional framework, and the advanced computational capabilities and dynamical properties of such complex systems.

Introduction

A neuron is composed of three main elements, a cell body (soma), dendritic trees and an axon. The dendritic trees are responsible for collecting the incoming electrical signals to the soma. Their number is typically greater than one and can exceed hundreds, while a single axon transmits the signal from the soma to the synapses of connected neurons. The diameter of the soma is a few tens of micrometers and is negligible comparing to the length of the dendrites and the axon, which can exceed millimeters, however, the soma is considered a crucial nonlinear computational element in the dynamics of the brain.

The long-lasting computational scheme, based on decades of experimental evidences is that a neuron functions similar to a single electrical excitable threshold unit (Fig. 1A). Additionally, it is well accepted that neurons sum the incoming electrical signals via their dendritic trees, and typically generate a spike to their axon if the membrane potential reaches a certain threshold, which varies among neurons (Fig. 1C, model I). According to this scheme the waveform of the spikes, e.g. rise time, peaks’ values, depolarization period and decay time to a resting potential, is consistently reproducible with high fidelity by the neuron, but varies among neurons.

A variety of theoretical models, based on the abovementioned scheme, were introduced during the last decades in order to describe the neuron as an excitable element. They vary between formal spiking neuronal models such as the leaky integrate-and-fire model and experimental evidence-based models such as the Hodgkin-Huxley model, which were followed by many combined variant models. Most of these models fairly capture the structure of observed biological spikes, but have difficulties in incorporating biological features, such as neuronal response failures in the intermittent phase38 and dynamical changes in the neuronal response latency, both mainly attributed to the dendrites. In addition, these standard neuronal models do not incorporate many nonlinear computations which are done in parallel processing and locally in each dendrite and its branches, including amplification of the synaptic inputs, local dendritic spike and coincidence detection. This new variety of dendritic computations leads to model a neuron similarly to a feedforward two-layer network with nonlinear hidden units, however, the output unit is typically a threshold unit, representing a single spiking element which transmits its signal along the axon. One can fairly conclude that the long-lasting computation scheme for a biological neuron consists of a single centralized excitable mechanism which linearly sums its entire incoming signals (Fig. 1C, model I).

In this work we present advanced scenarios for the computation scheme of a neuron, based on nonlinear and discontinuous responses by the dendrites and/or the neurons. The formulation of these scenarios requires to introduce the following three parameters: Th, W and f(W) (Fig. 1B); the parameter Th stands for a threshold for the generation of an evoked response in the neuron or its dendrite, the transmission function, f(W), of the incoming signal to the neuron, W, when W >Th, stands for a general continuous or discontinuous function (Fig. 1B, red lines).

The second scenario presented here is based on advanced dendritic computations, where the neuron sums its signals in a nonlinear manner. A signal from a dendrite is added to the summation only if it crosses a certain threshold, Thi, which varies among dendrites (Fig. 1C, model II). In both models (I and II in Fig. 1C) the neuron consists of a unique single central excitable mechanism. Based on new types of experiments we question this common scheme, and suggest that a neuron functions as an anisotropic threshold unit. More precisely, the neuron contains many independent excitable sites, each functioning as an independent threshold unit which sums up the incoming signals from a given limited spatial direction, most probably by a dendrite or a bunch of dendrites (Fig. 1C, model III). These anisotropic excitable sites are not identical and are characterized by different spike waveforms and different summation specifications. The neuron is a more complex and structured computational element than expected, and the implications on the functionality of neural networks are stimulating.

The mission of the proposed work demands the formation of a suitable experimental strategy which is based on the following fundamental steps and requirements. It initially demands stimulation of the neuron from several spatial directions, either independently or simultaneously. Indirect anisotropic stimulations of the neuron simultaneously require tunable stimulation timings on a sub-millisecond time scale. In addition, such stimulations schedule has to remain stable over timescales of many minutes while the neuronal responses have to be continuously recorded intracellularly. The achievements of all these requirements led us to implement anisotropic extracellular stimulations, which in addition have to be synchronized with the intracellular recording and stimulations. We indeed found some evident signatures in the responses of the neuron, which clearly differentiate between multiple stimulations from anisotropic sources and stimulations from a unique location. We have developed accordingly a set of experiments to reveal and to support the new proposed neuronal computational scheme.

Results

Experimental Setup

Our experimental results are based on a new available versatile setup, enabling complex multiple extracellular stimulations and recordings from a micro-electrode array (MEA), simultaneously with a patch-clamp stimulation and recording of a single neuron, selected from a cultured neural network (Figs 2A and B and Methods). Specifically, the in-vitro apparatus measurement (Fig. 2A) consists of an array of 60-electrodes with a diameter of 30 μm each, typically separated by 200 μm from each other (in a limited number of cultures separated by 500 μm, see Methods) and cover an area of (1.4 mm) X (1.4 mm) (Fig. 2A ) of the entire ~5 cm2 cortical tissue culture (gray circle in Fig. 2A ). The spontaneous spiking activity51 of the patched neuron as well as the nearby culture, sampled by the MEA, was typically silenced by the addition of synaptic blockers (Methods). Synchronized bursts activity was measured in the neuronal cultures before the addition of synaptic blockers. After the addition of synaptic blockers, no intra- or extra-cellular activity were observed over tens of minutes. In addition, repeated extracellular stimulations to the culture did not provoke cascades of neuronal responses (recorded extra- or intra- cellular). The stability of the neuronal response latency (Fig. 2E), much below a variance of a millisecond, also strongly excluded the possibility of leftover sparse connectivity in the culture. The stimulations and the recording of the intra- and the extra- cellular signals were done by two independent systems (Fig. 2B and Methods), and required a careful synchronization of their clocks. A sustained 20 μs matching between the two clocks was achieved using careful analysis of the relative drift of the two clocks and by using leader-laggard triggers for synchronization (Fig. 2B and Supplementary Fig. S1).

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Carboncopies Discussion

Randal Koene Thanks for posting! I hope I can read it soon. When I do, I'll comment with my impressions.

Diana Deca The paper is a lot of fun to read, almost like good poetry (to me at least, since it's really at the core of what I do). It does have a special style though. The experiments are very elegant, and I think the paper is really good both experimentally as well as conceptually. They had a 60 electrode rec/stim MEA in a cell culture while at the same time patching a neuron. They used the MEA electrodes to record and stimulate different parts of a neuron's dendritic tree while recording spikes intracellularly. Here are the experiments they did: 1. They stimulated the same neuron with 2 differently positioned electrodes and they got different spike waveforms. What you see in the figure is actually subthreshold and to me it makes sense the waveforms look different in that case. Differently shaped action potentials would have been weird and supporting their conclusion, which I did not really see. 2. They found that their neuron needs an 800mV stimulation for 2ms to get an action potential, so they put 500mVx2ms in two electrodes at the exact same time and did not get an action potential, suggesting nonlinear summation which again makes sense. They call it absence of spatial summation, which is not the case, it's just not linear. 3. They observed the same nonlinearity when doing intra- and extracellular stimulation. This is exactly the sort of experiments I think are vital before doing anymore recordings with neuroprosthetics, so now we are starting to have a basic idea of what happens to a neuron when stimulated by a MEA. As for the question whether this is relevant for human trials with neuroprosthetics, I think it is. There is another paper supporting that http://science.sciencemag.org/content/355/6331/eaaj1497 which shows that distal dendrites are 5 times more active than the soma of a neuron. Since dendrites make 90% of grey matter, this is definitely important. All of this might actually suggest we should not even try to have single neuron resolution with neuroprosthetics, since the neuropil signals are much stronger than what we considered to be the somatic ones. It's definitely an important step forward conceptuall for WBE, figuring out what exactly it is we are stimulating and measuring from.

Dendrites are more active than expected. Dendrites occupy more than 90% of neuronal tissue.…
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Gregory Norris Interesting article. This definitely helps clear the muddy waters between the cases for base-component emulation and network signal emulation.

Randal Koene (Diana Deca) I finally got around to reading this paper on neurons as multiple independent threshold units. My impression:

Obviously, this isn't the first paper to point out that neurons are not quite like the simplified approximations in artificial neural networks, and that we have to take care to take into account their unique arrays of interacting dendritic input sites with their particular response functions. This is often referred to as (Nonlinear) Dendritic Computation and has an academic literature spanning at least a decade.

Sardi et al is a good experimental methods paper and take good care (although I don't think they plumbed the depths of combinatorial possibilities or techniques such as sensitivity analysis).

I think there is a cautionary note to sound, because the paper might lead some to draw the wrong conclusions. And I think there is an important insight about the difference between data and model to highlight based on the reported operation of the neuron.

Cautionary note: Dendritic site-specific unique behavior does not invalidate system identification methods based on predicting spatio-temporal spike activity at populations of neurons. Some care needs to be taken to not assume model functions that are unable to capture the necessary computational complexity, but in principle you can still capture the effects of input at each one of the dendritic sites and their interactions by observing the spike times of the 'ouput' and 'input' neurons. To capture well and predict reliably you do need to have a lot of input-output spike data.

Insight: Creating a functional model from structural image data only (e.g. high resolution connectome image stacks) takes more than counting connections or source-destination neuron addresses. You need to know quite a bit about dendritic locations, channel types, etc. A translation library might be able to use stochastic rules about dendritic response functions typically found at given locations between specific neuron types in a specific region of the brain. Before you have a library that detailed and a way to finesse the selection of parameters from the proposed probability distributions in a manner that results in a working whole at the system level, it is probably a good idea to keep on recording activity (in addition to any connectome information).

RE Diana Deca Indeed, dendritic computation is old, almost as old as neuroscience and the first patch clamp recording if not even ramon y cajal, so I am happy to be in this subfield. I definitely agree just reading the text is misleading, as what they discover is nonlinearity not absence of spatial summation. I do know that nature makes the authors inflate their findings to make them sound shocking, which is what a neuron not behaving like a neuron would be.

Randal Koene (Btw, Diana Deca) I think the question of 'resolution' in a neural prosthesis is almost entirely disconnected from the number of neurons, dendrites, etc, within the region being replaced. The performance metric has more to do with the interaction of the prosthesis with the surrounding tissue. For example, if you replace a hippocampal region with a prosthesis, does it provide the necessary function that the remaining regions expect, and can it interact with them in the way desired. Being able to do that may require that the prosthesis be able to detect input activity on anywhere between 0-100% of the replaced neurons that were receiving input from elsewhere, and may require that it be able to produce output action potentials on anywhere between 0-100% of the replaced neurons that were sending output to elsewhere. Those proportions remain to be empirically determined, as do many other metrics for a good neural prosthesis. But hopefully, as long as the prosthesis is fully functional on its I/O then we are free to be creative and adventurous with the inside of the black box. ;-)

Randal Koene I propose that the metric for the scale of a prosthesis be simply (a) the number of parameters that need to be set/learned for whatever functions are being used within the prosthetic model, and (b) the number of I/O sites of the implant. E.g. you could end up with something like: The new CA3-CA1 prosthesis CA3CA1v1.0 boasts 97% capture of original system behavior with 96.5% predictive reliability on 100 hours of test data, using 1.2GigaParameters and 2 million sites arranged homogeneously. :-)

[P.S. For this imaginary v1.0 product, I am not suggesting that homogeneous recording/stimulation site distribution is optimal, but it is easier given that it is also hard to position a new implant, and wasting a few excess sites on overlap may not be a high price to pay.]

RE Diana Deca Sorry for the truncated message I am literally sticking two electrodes into a fly brain. While it is true that performance (eg: behavioral output) is a good parameter for seeing if a neuroprosthetic is working or not, it is also important to understand what exactly they are stimulating and how much. While the text of the paper may be a bit inflated and nonlinear summation is not exactly shocking, their data nevertheless shows that a single neuron within an implant needs around 800mV to generate an action potential of around 100mV and that 2x500mV is not enough. Also, sub and intracellular stimulation do not add up. What is relevant here for prosthetics is I would say that dendrites have 5 times more spikes than the soma, so when we do single unit recordings we’re actually recording mostly the dendritic signal not the somatic all or none action potentials as many people initially thought. Of course with multi unit recordings it kind of averages things out, but it challenges the age old idea that we are recording action potentials with some noise with implants and that might give some old professors a heart attack. If resolution is the number of microns a single electrode records from, then we just need to know approximately just how much of that is cables (dendrites) and how much is a hub (soma)

RE Randal Koene (Diana Deca) while a totally appreciate your professional insight - and I certainly don't want to keep interrupting your fly brain recording - I think we need to drill down a little bit on what you're trying to say here. It's very easy for this to be confusing conversation where it doesn't need to be. Let me just make a few statements, and you can tell me if you agree or not, and if not, why not:

1. If a prosthetic can produce the expected output (axonal discharge) that causes the desired responses in a target region (e.g. in CA1) then the prosthetic is operating correctly.
2. If a prosthetic model can generate stimulation (e.g. in CA1) such that the target neurons will fire when expected then this is equivalent to such correct axonal discharge. (There is a side-issue that placing electrodes in CA1 with the intent to cause firing of a given neuron is likely to cause ancillary charges elsewhere, because they are not generally intracellular electrodes... to determine if this matters, given the coding the brain uses, remains a topic of study.)
3. If we place recording sites for a prosthesis in a source location (e.g. CA3) we will record changes in (normally extracellular) potential, which is representative of an amalgam of somatic and dendritic activity.
4. Given a sufficient number of recording sites in the source and the destination (e.g. CA3 and CA1), we may discern enough different spatio-temporal patterns of activity to deduce functions representing how the source affects the destination.

I don't think we're necessarily disagreeing. Whether you look at it in terms of which dendritic site activates or which source neuron is sending activity to a dendritic site, you still need to take into account non-linear combinations of activity to correctly predict when the destination neuron will fire.

And as I tried to point out in points 1-5, when you're building a neural prosthetic, you try to record enough hints to make out predictive functions, but you're generally not sure what precisely you are recording from anyway (somatic, dentritic, otherwise). It's local field changes in some vicinity of the electrode. (Almost like some nondescript 'voxel' as in MRI.) Similarly when you stimulation. You are pushing charge into some small region of the tissue, not specifically into a particular neuron, but you hope elicit meaningful response spikes.

It is theoretically possible to build implants that patch-clamp neurons, but that adds a serious layer of technical difficulty.

Interestingly (!) some of the uncertainty, mushiness and imprecision goes away as you start prosthetically replacing more (adjacent) pieces (right up to whole brain emulation). After all, if not just your neurons in CA3 have been replaced by prosthesis, but the ones in CA1 as well, then you can mathematically insure that exactly the right target responses are elicited in the CA1 prosthesis as a result of function output from the CA3 prosthesis. Another reason to go for the whole thing and do whole brain emulation. ;-)

RE Randal Koene (Diana Deca) Ugh, I can't believe I'm interrupting more, but rereading your last sentence ('we just need to know approximately just how much of that is cables (dendrites) and how much is a hub (soma)') I couldn't help wondering: If you were told that 80% of what you are recording with an array of some resolution is dendritic, how would you use that information in the prosthetic model?

RE Diana Deca (Randal Koene) if the output is a voltage recorded on specific electrodes of the MEA then I fully agree, I guess I thought you meant behavioral not electrical output. While in some sense we could abstract from single neuron issues as long as the electrical output is the same, the predictive value of the model for that stimulation will be affected as long as the input output measures are only correlational and not causal. One compromise could be looking at the slices-and trying to get an estimate of morphology parameters, but i still think this would provide only weak explanatory power to the model. Sorry if this sounds blunt I think we’re almost on the same page

RE Diana Deca (Randal Koene) we could model that as cable properties rather than the on/off switch that is the soma theoretically. It would depend on morphology too for ex basal dendrites are more linear than tuft ones

RE Diana Deca Very good point about this being less of an issue as WBE progresses! The question is how much info do we need to start one

RE Randal Koene I'm getting a bit lost I think. I don't understand what you mean above about 'correlational' vs 'causal'. Let's try a very simple example. Let's assume for a moment that CA3 contains only 3 neurons and CA1 contains only 3 neurons. Let's also acknowledge that the connections between CA3{1,2,3} and CA1{1,2,3} are through dendritic sites that interact and operate non-linearly on the activity of the target neuron (CA1{1,2,3}).
Now, If put an electrode with 4 recording sites into CA3 and an electrode with 4 recording sites into CA1:

1. a) The recording sites will not line up directly with any particular neuron, but will provide "coverage" of some kind.
2. b) As we record changes at the 4 CA3 and the 4 CA1 electrode sites, we build up a pile of data that we can use to train/regress a model that represents how the activity in CA3 predicts activity in CA1.
3. c) We can then stimulate through 4 stimulation sites (let's say they are the same as the recording sites) in CA1. Those won't precisely stimulate any particular neuron (or dendrite), but again provide coverage of some sort.
4. d) We may even attempt to use some sort of smart stimulation to be more precise (like a spanish hat with positive in the middle and negative in the ring around it)... which is hard to imagine in this tiny example.
5. e) For our performance test, we can measure the degree to which our stimulation seems to result in the desired activity for known test sets of CA3-CA1 data. I admit this won't work in the super simple setup here... the stimulation totally obscures and wipes out recording of the response. It would be better to test this with at least 3 recording layers. Otherwise, behavior is the only useful test.

All the above, just to be super concrete. Now, in this, what did you mean would be correlation vs causation?

I might actually be starting to see where we are talking past each other. It sounds like you are thinking of this in terms of an attempt to use neuroscientific models of neurons (at some resolution, eg cable models), whereas I'm thinking purely in terms of building a working device with max performance.

My stance is this: It might be that a particular neural model is a good set of functions to use to approximate function, but it might also get in the way, because it places particular demands on what must be known (e.g. connectivity), and if that information is missing or partial then the results could be worse than a black box model with more easily trainable parameters.

I'm not against neuron models per se, but I'm for being really really open in our choice of functions to use in approximating system behavior.

Keeping in mind that when we do recording, we may not actually be obtaining information that relates directly to any particular neuron model (will we see dendritic sizes? will we see connection locations and receptor types? will we record activity for which we know if it comes from a dendrite?).

If the information we can actually collect and use to generate our prosthetic model is more 'voxel'-like in nature (as it typically is right now in array recordings) then we may as well be honest and declare that this is the knowledge we have about the system and that we wish to use to train the model.

RE Diana Deca True, unless the predictive value of the model is affected and we get different results everytime, then it will be useful to add information about what exactly is in those voxels

RE Randal Koene (Diana Deca) And then... you would actually have to collect that information as well, which makes the data collection so much harder, so I really hope not :-)

RE Randal Koene Btw, I'm assuming when you say "predictive value of the model is affected" you mean affected by our ability to feasibly (in non-infinite time) train the model parameters with the information recorded? The computational difficulty is certainly a hurdle on top of the recording problem. This is what Brian Robinson was running into with the plasticity part of the hippocampal model. I agree, if _deducing_ functions that predict well becomes computationally intractable then you need to dig deeper and make your black boxes smaller (switch to generating predictive functions for somas or patches of dendrite) instead of for collections of neurons. Yup. There is always this balance and trade-off... harder recording to solve computational intractability.

Diana Deca That’s basically the WBE by SID approach, which is I think the best tool there is for now. That’s why I was suggesting in my book chapter to use these models in hardware implementations like for grid cells and robot navigation, because then it’s easier to check if the emulation is at least efficient if not a perfect copy.

RE Randal Koene On a side-note... what names/labels could we use for other approaches to WBE?
For example, if a WBE (or prosthesis) is attempted by using image data to identify actual bits of connectome and then use a library of probabilistic response characteristics to convert that to a functional representation, or at least use the images as a way to constrain possible parameter values in a SID model... would that be called a "Constrained SID (CSID)" approach, or even a "Connectome Constrained SID (CCSID)"?

According to Wikipedia: "The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data." Hm, this doesn't seem to exclude data beyond activity data, although I think it's only been used as such.

Also... abbreviations seem to vary. I've seen it abbreviated as SI, and also as SYSID, and now SID.

RE Diana Deca (Randal Koene) I think Shawn Mikula might be able to answer that