How to go about reproducing a conductance-based model and troubleshooting it

Standard

Building a model from a publication

All the equations are there, you’ve got figures of model voltage traces that show what the model is supposed to look like, and you’ve got a table of parameters that the authors used to get those figures. It might seem like a breeze to type all that up in Matlab and pop out identical results in reproducing that conductance-based model, but your patience and methodicalness will be rewarded. I recommend starting with the function code that will be called by your ode solver and building in one conductance at a time (see example below).

function dy = myNeuron(t,y)
Vm = y(1); % mV
H = y(2); % gating variable for a conductance
M = y(3);

cm = 1; % nF
g_leak = 0.001; % microS
g_k = 0.2;
g_Na = 0.3;
v_leak = -40; % mV
v_k = -80;
v_Na = 100;

h_inf = 1/(1 + exp((Vm + 15)/6)); % hypothetical steady state gating
m_inf = 0.5*(1 + tanh((Vm - 31)/22));

I_leak = g_leak*(Vm - v_leak); %nA
I_K = g_k*H*(Vm - v_k);
I_Na = g_Na*M*(Vm - v_Na);

dy(1) = (-I_leak - I_Na - I_K)/cm;
dy(2) = (h_inf - H)/tau_h;
dy(3) = (m_inf - M)/tau_m;

This could be your inner function, myNeuron, that is called by the ode solver you choose.

[T,Y] = ode45(@(t,y) myNeuron(t,y),[0 1],y0);

Start with the leak conductance alone – making the most boring, passive neuron ever.

Make sure that no matter what your initial voltage is, the voltage trace that you simulate will relax to the resting voltage you’ve chosen. Does it look like it’s taking too long to get there? Or getting there too quickly? Now is a great time to check your units and the time scale or time step you’re using. If anything about this first step of simulating a passive neuron got weird:

• Make sure the units check out. Sometimes authors publish their parameters in terms of conventional units but the units wouldn’t check out on both sides of the equation. For example, without putting a 1e-3 in front of the nA. Below is my little cheatsheet of unit conversions.
• Check the implementation of the ode solver you’re using. Make sure the variables haven’t gotten switched, and make sure they’re being assigned to the solver’s channels. Matlab documentation is helpful here: Example of Implementing ODE Solver and Parametrizing Matlab Functions.
 unit conversion Voltage (Volt) Current * Time / Capacitance (Ampere * second / Farad) Current (Ampere) Charge / Time(Coulomb / second) Capacitance (Farad) Charge / Voltage(Coulomb / Volt) Conductance (Siemen) 1 / Resistance = Current / Voltage(1 / Ohm) = (Ampere / Volt)
 giga 1e+09 mega 1e+06 kilo 1000 unit 1 centi 0.01 milli 0.001 micro 1e-06 nano 1e-09 pico 1e-12

If you’ve got a working passive neuron with only a leak conductance, you can start building in the rest of your conductances. I recommend dividing your conductances into “major” and “minor” conductances. The major currents are leak, one main inward, and one main outward current. Work with those first to make sure you get something reasonable that qualitatively resembles a less complex model with only “major” conductances. For example, the Hodgkin-Huxley model only has leak, sodium (Na+), and potassium (K) conductances; or the Morris-Lecar model which has only leak, calcium (Ca), and potassium (K) conductances. Then add in your other currents one at a time and observe that they are changing the model behavior in a way that is expected. For example, including another K conductance should slow down spiking or eliminate it while adding in a Ca conductance should make a more excited response or produce some slow-wave bursting dynamics.

Troubleshooting a complete model

If you’ve gone through the above steps, or if you came to this guide with a complete model that isn’t working. Here are my tips for fixing it.

• Stupid things like typos are common. It might not even be your typo but the author’s typo. If possible, check the paper you’re using against another paper that uses the same model. Look for any discrepancies. Generally, the conductance equations should be exactly the same. It’s rare that those change from paper to paper. The parameters would be different from paper to paper (or even within the same paper), but make sure they look reasonable and use comparable units (i.e. it would be weird if one paper used 0.1 picoS for gNa but another paper used 100 mS since the difference is several orders of magnitude and it’s unlikely that both models are functional. Usually one of those is a mistake).
• The ode solver may not be right for the model. If you have lots of dynamical variables (>3), the likelihood of stiffness might go up since that can happen with variables that change at very different time scales. One way around this is to choose a very tiny time step while troubleshooting. A tiny time step will eliminate the problem of stiffness for any ode solver you use, so even though it will take much longer to simulate a short run, you can rule out other stuff in the meantime.
• Or the simulation time scale is too coarse in general and you’d need to simulate on a finer timescale anyway even if you’ve chosen the best ode solver for your system.
• Go through each conductance and do a sign check. Make sure that the conductance produces a current of the correct sign and has qualitatively appropriate dynamics (fast, slow, etc.). Do this with the other currents off. Make sure the current is changing the voltage in a way that seems reasonable/expected (i.e. an inward current should depolarize your voltage).
• Look for interdependencies in the conductance equations. If there are any, test those conductances together and make sure one isn’t causing the other to be dysfunctional.

If all else fails, I’ve had success with simply starting from scratch with clean code (no copy/pasting sections of the old code). I know that seems daunting, but you’ll wish you had done it sooner if it solves your problem. I’ve done this before and never ever found the mistake in my original code even though the new code actually worked – so sometimes a small error is really hard to find even when you know it’s in there somewhere.

Crab stomatogastric ganglion dissection guides

Standard

When I first started in the Marder Lab as a graduate student, I had the good fortune of being trained in the crab stomatogastric ganglion dissection by a very patient and helpful senior grad student. I also got to watch several other experienced dissectors perform this dissection. This allowed me to come up with my own customized dissection protocol that combined the best aspects of the varied techniques I saw. I wanted to document my dissection protocol in the hopes of helping others who might be trying to learn this dissection, so I made a two-part illustrated guide. This ended up being incorporated into my first official publication in the Journal of Visualized Experiments (Gutierrez and Grashow, 2009).

These guides are still available on the resource page of the Marder lab website, but I’ve made them available here as well.

Gutierrez GJ, Grashow RG (2009). Cancer borealis stomatogastric nervous system dissection. J Vis Exp. Mar 23(25). pii: 1207.

“You can’t be what you can’t see”

Standard

I’m Snapchatting snippets of my work day to give the STEM-curious an inside view of what it’s like to do science. My goal isn’t to teach or explain my research in great detail. That would be impractical given the nature of Snapchat in which posts are really short clips or pictures and expire after 24 hours. My goal is to make my job relatable and less intimidating to anyone interested in science but unsure about whether they can see themselves doing it as a career.

This past week has been fun experimenting with this project, but it’s actually a lot more challenging than I expected. I’m a really private person and I’m not one to pull out my phone and take a random selfie, so I’ve been much more shy than I’d like to be about talking to my phone when my colleagues are around – but I’m trying to be bolder. The other challenge I’m having is with what to show. I do all kinds of things on any given day, but it can be tricky to say something snappy about what I’m working on without much context. That’s where I could use your help. Check out my Snapchat Stories and give me your suggestions for what to present. Also, please share my SnapCode with any young people you know who might be interested in seeing examples of women in science at work. Oh, and one more thing. If you’re doing a job where role models like you are lacking, try doing your own Snapchat thing. We can do it together!

E/I balance rescues the decoded representation that is corrupted by adaptation

Standard

Gabrielle J. Gutierrez and Sophie Deneve

Spike-frequency adaptation is part of an efficient code, but how do neural networks deal with the adverse effects on the encoded representations they produce? We use a normative framework to resolve this paradox.

Fig. 1: Adaptation shifts response curve. The shift in neural responses maintains a constant response range for an equivalent area under the stimulus PD curve.

The range of firing rates that a neuron can maintain is limited by biophysical constraints and available metabolic resources. Yet, neurons have to represent inputs whose strength varies by orders of magnitude. Early work by Barlow1 and Laughlin2 hypothesized and demonstrated that sensory neurons in early processing centers adapt their response gain as a function of recent input properties (Fig. 1). This work was instrumental in uncovering a principle of neural encoding in which adapting neural responses maximize information transfer. However, the natural follow-up question concerns the decoding of neural responses after they’ve been subject to adaptation. There’s no question that this kind of adaptation has to result in profound changes to the mapping of neural responses to stimuli3,4 – so how are adapting neural responses interpreted by downstream areas?

By using a normative approach to build a neural network, we show that adapted neural activity can be accurately decoded by a fixed readout unit. This doesn’t require any synaptic plasticity – or re-weighting of the synaptic weights. What it does require, as we’ll show, is a recurrent synaptic structure that promotes E/I balance.

Our approach rests on the premise that nothing is known from the outset about the structure of the network. All we known is the input/output transformation that the network performs. For this study, that I/O function is simply a linear integration of feedforward input the network receives. Given some input, c(t), we expect some output, x(t), such that $\dot{x}(t) = Ax(t) + c(t)$. The variable, x(t), is called the target signal because it is what we expect the network to produce given the input, but what the network actually puts out is denoted as x̂(t). We assume that the true network output is a linear sum of the activity of the network units, $\hat{x}(t) = \sum_n w_n r_n(t)$, where ri(t) is the activity of neuron i and wi is its readout weight. It is this actual network output, x̂(t), that will be compared to the target output, x(t).

With these assumptions, we set up an objective function, E, to be minimized. We want to minimize the representation error of the network as well as the overall neural activity. In other words, we derive a network that from the outset has the imperative to be as accurate as possible while also being efficient. The representation error is the squared difference between the decoded estimate that’s read out from the network,, and the output signal we should expect, x, given the input. The metabolic cost is a quadratic penalty on the network firing activity of all n neurons. So the objective looks like this: $E(t) = [x(t) - \hat{x}(t)]^2 + \mu \sum_n r_n(t)^2$

To derive a voltage equation from this objective (see these notes for detailed derivation), we rely on the greedy minimization approach from Boerlin, et al 5, which involves setting up an inequality between the objective expression that results when a neuron in the network spikes versus when no spike is fired in the network: E(t|no spike) > E(t|spike). This forces spikes to be informative. A spike may fire only if the objective is minimized by that spike. A spike must make the representation error lower than if a spike were not to have been fired at that time step.

Fig. 2: Spike-frequency adaptation. A history dependent spiking threshold (green) increases and decays with each spike fired (blue) in response to a constant stimulus (pink).

Knowing that the voltage of a spiking neuron needs to cross a threshold before a spike is fired, we let this inequality represent that concept so that after some algebra, the left-hand side expression is taken to be the voltage and the right-hand side is the spiking threshold. In other words, V > threshold is the condition for spiking. Therefore, $V_i = w_i(x - \hat{x}) > \frac{w_i^2 + \mu}{2} + \mu r_i = threshold$.

Let’s first take a look at the spiking threshold. Notice how it is a function of the neuron activity variable, r(t). This means we’ve derived a dynamic spiking threshold that increases as a function of past spiking activity (Fig. 2). Thus, spike-frequency adaptation fell into our lap from first principles. The dynamic part of this threshold is a direct result of the metabolic cost that was included in the objective function.

Fig. 3: Schematic of derived network.

Taking the derivative of the voltage expression gives us an equation where each term can be interpreted as a current source to the neuron. The resulting network is diagrammed in Figure 3 where you’ll see that the input weight to a given neuron is the same as its readout weight and proportional to the recurrent weights it receives as well as its own self-connection (i.e. autapse). Because our optimization procedure didn’t specify values for these weights – just the relationships between them – the weight parameter, wi, for any given neuron i is a free parameter. But the value of that parameter has consequences for the adaptation properties of the neuron in question (Fig. 4).

Fig. 4: Adaptation profiles for heterogeneous neurons. The weight parameter determines how excitable a neuron is and its time constant of adaptation.

Neurons with a large weight not only have higher baseline firing thresholds than their small weight counterparts, they have stronger self-inhibition. In contrast, small weight neurons are intrinsically closer to threshold, so they have a higher firing frequency out of the gate, but they burn out quickly because of spike-frequency adaptation. From here on, I’ll refer to the neurons with a small weight as excitable and the large weight neurons as mellow. These heterogeneous adaptation profiles have an important role to play in the network we’ve derived.

Fig. 5: Network response to a stimulus pulse. Neurons fire in response to the stimulus (top, raster) with the most excitable neurons firing first (light green) and the mellower neurons pitching in later (dark blue). Despite time-varying activity in the individual neurons, the network output (orange) tracks the target signal (grey).

To illustrate how this panoply of diverse neurons work together to represent a stimulus, take a look at Figure 5 in which a pulse stimulus has been presented to the network. For the duration of the pulse, the network as a whole does a great job of tracking the stimulus, forming a stable representation over that time. Any single neuron individually does not maintain a stable representation of the stimulus, but the network neurons coordinate as an ensemble. The excitable neurons are the first responders, valiantly taking on the early part of the representation. But they quickly become fatigued. That’s when the mellow neurons kick in to take up the slack. This coordinated effort is all thanks to the recurrent connectivity. When a neuron is firing, it is simultaneously inhibiting other neurons, basically informing other neurons that the stimulus has been accounted for and reported to the readout. But when adaptation starts to fatigue that neuron, it dis-inhibits the other neurons. At some point the amount of input that is going unrepresented outweighs the amount of inhibition coming from the active neuron, causing a mellower neuron to be recruited in carrying the stimulus representation.

Fig. 6: E/I balanced currents reduce error. Left, excitatory and inhibitory currents impinging on an example neuron in response to three different stimulus presentations. The neuron in the top plot belongs to a network with random recurrent connections that are not E/I balanced. In the bottom plot, that neuron is part of an E/I balanced network. Right, the representation error for the unbalanced network (grey) is higher than for the balanced network (black).

This connectivity scheme is inherently E/I balanced, meaning that excitatory currents to an individual neuron are closely tracked to the inhibitory currents entering that same neuron (as shown in the left panel in Fig. 6). When the network takes on a random recurrent structure, even though the currents are somewhat balanced over a long time, they aren’t as tightly balanced as in the recurrent connectivity structure that we derived. The balanced recurrent connectivity scheme is also what’s keeping things accurate (Fig. 6, right plot). In fact, the connectivity structure is entirely derived from the error term in the objective.

Now that we have a model with adaptation and E/I balanced connectivity, we use it to model a network that encodes orientation, such as in area V1 in visual cortex. To do this, we made a neural network with two cell types: mellow and excitable.

Fig. 7: Schematic of orientation coding network. Each orientation is represented by a pair of neurons, one excitable and one mellow neuron. Only a few connections coming from the outlined neuron are shown. Inhibitory connections terminate in a bar and excitatory connections terminate in a prong.

Each neuron has a preferred orientation which is set by the complement of input weights it receives. The preferred orientation of each mellow neuron overlaps with the preference of one other excitable neuron. That means that each orientation is preferred by a pair of network neurons, one excitable and one mellow (Fig. 7). It’s worth pointing out how the derived connectivity interacts with neuron preferences. Specifically, neurons with similar preferences inhibit each other most strongly, whereas neurons with opposing preferences excite each other. This seems counterintuitive – and even contrary to the experimental data – but it reflects the effective encoding strategy at work here. Neurons with similar preferences are competing with each other for the chance to report the stimulus to the readout. If all of the neurons reported at once, the readout would be overwhelmed and unable to decode the stimulus as accurately because the representation would too often reflect the intrinsic properties of the active neurons. On the other hand, neurons with opposite preferences can afford to excite each other because it’s almost like a game of chicken. The active neuron is betting that the opposing neuron isn’t receiving strong input and can therefore feel confident that exciting that neuron won’t be enough to bring it to spike. This set up keeps all neurons relatively close to their baseline spiking thresholds so that any given neuron is ready to be recruited at the drop of a hat.

Fig. 8: Tuning curves. The excitable neurons have broader tuning curves (light green) than the mellow neurons (dark blue).

The tuning curves for the excitable and mellow subpopulations reveal their particular characteristics (Fig. 8). Excitable neurons have a broader tuning curve than their mellow counterparts. Their tuning curves are also higher magnitude than the mellow ones, but both tuning curves were normalized to unity in the figure. These tuning curves represent the early responses of the network neurons to a series of stimulus presentations. By comparing them to the late part of the response to those same stimuli, we can see how the tuning curves change to accommodate the effects of adaptation (Fig. 9). The tuning curve for the late responses in the excitable neurons shows a decrease in the amplitude of the curve near the preferred orientation (left, Fig. 9). This is what most people would expect to see as a result of adaptation. However, the situation for the mellow neurons is counter to those expectations (right, Fig. 9). Their late responses show an increase in activity at the preferred orientation. This is because the excitable neurons are adapted more strongly than the mellow neurons, which means that the mellow neurons have to pitch in to save the representation after the excitable neurons burn out. Thus the mellow neurons tuning curves are facilitated due to adaptation, not suppressed.

Fig. 9: Tuning curves change after adaptation. Tuning curves for early responses as shown in Fig.8 are in grey. After adaptation, the tuning curves are suppressed for the excitable neurons (left, light green), but facilitated for the mellow neurons (right, dark blue).

We showed that E/I balance works hand-in-hand with adaptation to produce a representation that is both efficient and accurate. Sure, we could’ve allowed adaptation to result in a perceptual bias. Our model doesn’t exclude that possibility, but we paid particular attention to the short-term effects of adaptation, and to the subtle changes that adaptation produces in neuron tuning without degrading the network’s ability to accurately encode the stimulus. The bigger picture here is that variability is part of the optimal solution rather than a problem.

Fig. 10: Variability in network neuron responses. The spike rasters from the network are color coded for each stimulus presentation. The stimulus was identical across trials but preceded by a different randomized stimulus sequence. Individual neuron rasters are organized horizontally so that each line represents the spikes from a given neuron.

We illustrate that principle with the overlaid spike rasters in Figure 10 in which the network is presented with the same stimulus on three separate occasions. The only difference between those presentations are the randomized stimulus sequences presented before each one. The history dependence of spike-frequency adaptation produces highly variable neuron responses to the same stimulus over different trials. Despite that variability in the spike timing and firing rate of individual neurons, the network output is very accurate across those three presentations of the stimulus. Adaptation is the catalyst for the redistribution of spikes, while E/I balance is the means by which spiking activity is redistributed in a manner that will preserve the representation. With adaptation enforcing an efficient encoding and E/I balance maintaining an accurate representation, the network can have its cake and eat it too.

1. Barlow, H. B. Reconstructing the visual image in space and time. Nature 279, 189–190 (1979).
2. Laughlin, S. A Simple Coding Procedure Enhances a Neurons Information Capacity. Z. Naturforsch., C, Biosci. 36, 910–912 (1981).
3. Series, P., Stocker, A. A. & Simoncelli, E. P. Is the Homunculus ‘Aware’ of Sensory Adaptation? Neural Comput 21, 3271–3304 (2009).
4. Solomon, S. G. & Kohn, A. Moving Sensory Adaptation beyond Suppressive Effects in Single Neurons. Current Biology 24, R1012–R1022 (2014).
5. Boerlin, M., Machens, C. K. & Denève, S. Predictive Coding of Dynamical Variables in Balanced Spiking Networks. PLoS Comput Biol 9, e1003258–16 (2013).

Standard

During one of my visits to a Girls Who Code group, one of the students asked me what advice I have for the next generation of girls. Over the years, I’ve been lucky enough to get some really good advice. So I thought I’d pass it on and also share some things I’ve learned myself.

This was advice given to me by my amazing PhD advisor, Eve Marder. If you don’t make your own mistakes and you let someone else make them for you, you will become “bitter and twisted”. Mistakes are OK, they’re part of being human and part of our learning process. But if you let others dictate your path and allow them to make choices for you, there is nothing to learn. Don’t rob yourself of the personal growth that comes from holding yourself accountable for your choices – even if you’re wrong. Make your own mistakes and make them with courage and conviction!

Value your personal talents

Sometimes you’ll be tempted to assume that what comes easy to you is easy for everyone. Don’t overlook your own talents simply because you have to expend minimal effort to pull them off. It’s especially easy to neglect the things you’re good at when you don’t receive enough encouragement or recognition for them, so when someone complements you for a job well done, don’t write it off as a fluke. You might have a gift that is worth nurturing.

Fake it till you make it

I’ve heard this many times, from many people, and it never stops being good advice. At every stage of your life and career you’ll find yourself doubting your own abilities. It doesn’t help that there will be people who will help to seed that doubt (most without even realizing it). You’re not alone in thinking that you’re not qualified enough, or smart enough for whatever it is you deserve a chance at – it happens to the best. In those times, all you can do is fake it until you convince yourself. If you push on, there will come a point where you realize that you’re drawing on real knowledge and brainpower to “fake” your way through a situation. Confidence can be worn like a coat, and you shouldn’t leave home without it.

Don’t take any opportunity for granted

Sometimes you’ll want to coast through a task because you’re just doing it for your college applications or to check some box somewhere. Other times, you’ll wish you could walk away from an insurmountable challenge or you might be too intimidated to even try in the first place. Yet these are all opportunities to do something awesome, to learn about yourself and the world, and to gain skills or knowledge that you didn’t have before. Don’t take them for granted. You’ll be better off if you make the most out of the experiences and the challenges you take on, so go all the way!

Standard

The goal of this site is to track the speaker composition of conferences in neuroscience, particularly with respect to gender representation. The progress of science benefits from diverse voices and ideas. Conference panels that are diverse with respect to gender, race, ethnicity and national origin help advance this goal. Homogenous conference programs are generally not representing their field, missing out on important scientific findings, and are one important factor contributing to the “brain-drain” of talented female and minority scientists from the scientific workforce. As a group, BiasWatchNeuro has formed to encourage conference organizers to make every effort to compose programs that incorporate diverse panels.

Bias is often unconscious and unintended. Indeed, most of us are biased, but with appropriate awareness, many people are now successful at overcoming their biases. The purpose of this site is to provide data and other resources to facilitate that effort, and in particular, to raise awareness of any gender bias in the selection of conference speakers, so that these disparities can be addressed. See “how can I help” for more information.

Send information about conferences, seminar series or other scientific programs to biaswatchneuro@gmail.com