The probability of life

Estimates from the Kepler satellite suggest that there could be at least 40 billion exoplanets capable of supporting life in our galaxy alone. Given that there are perhaps 2 trillion observable galaxies, that amounts to a lot of places where life could exist. And this is only counting biochemical life as we know it. There could also be non-biochemical lifeforms that we can’t even imagine. I chatted with mathematician Morris Hirsch a long ago in Berkeley and he whimsically suggested that there could be creatures living in astrophysical plasmas, which are highly nonlinear. So let’s be generous and say that there are 10^{12} planets for biochemical life to exist in the milky way and 10^{24} total in the observable universe.

Now, does this mean that extraterrestrial life is very likely? If you listen to most astronomers and astrobiologists these days, it would seem that life is guaranteed to be out there and we just need to build a big enough telescope to find it. There are several missions in the works to detect signatures of life like methane or oxygen in the atmosphere of an exoplanet. However, the likelihood of life outside of the earth is predicated on the probability of forming life anywhere and we have no idea what that number is. Although it only took about a billion years for life to form on earth that does not really gives us any information for how likely it will form elsewhere.

Here is a simple example to illustrate how life could grow exponentially fast after it forms but take an arbitrarily long time to form. Suppose the biomass of life on a planet, x, obeys the simple equation

\frac{dx}{dt} = -x(a-x) + \eta(t)

where \eta is a zero mean stochastic forcing with variance D. The deterministic equation has two fixed points, a stable one at x = 0 and an unstable one at x = a. Thus as long as x is smaller than a life will never form but as soon as x exceeds a it will grow (super-exponentially), which will then be damped by nonlinear processes that I don’t consider. We can rewrite this problem as

\frac{dx}{dt} = -\partial_x U(x) + \eta(t)

where U = a x^2/2 - x^3/3.


The probability of life is then given by the probability of escape from the well U(x) given noisy forcing (thermal bath) provided by \eta. By Kramer’s escape rate (which you can look up or I’ll derive in a future post), the rate of escape is given approximately by e^{-E/D}, where E is the well depth, which is given by a^3/6. Thus the probability of life is exponentially damped by a factor of a^3/6D. Given that we know nothing about a or D the probability of life could be anything. For example, if we arbitrarily assign a = 10^{10} and D = 10^{-10}, we get a rate (or probability if we normalize correctly) for life to be on the order of e^{-10^{40}/6}, which is very small indeed and makes life very unlikely in the universe.

Now, how could it be that there is any life in the universe at all if it had such a low probability to form at all. Well, there is no reason that there could have been lots of universes, which is what string theory and cosmology now predict. Maybe it took 10^{100} universes to exist before life formed. I’m not saying that there is no one out there, I’m only saying that an N of one does not give us much information about how likely that is.


Addendum, 2019-04-07: As was pointed out in the comments the model as is allows for negative biomass.  This can be corrected by adding an infinite barrier at zero (i.e. restricting x to always be positive) and this won’t affect the result.  Depending on the barrier height and noise amplitude it can take an arbitrarily long time to escape.

New paper in Cell

 2018 Dec 10. pii: S0092-8674(18)31518-6. doi: 10.1016/j.cell.2018.11.026. [Epub ahead of print]

Intrinsic Dynamics of a Human Gene Reveal the Basis of Expression Heterogeneity.


Transcriptional regulation in metazoans occurs through long-range genomic contacts between enhancers and promoters, and most genes are transcribed in episodic “bursts” of RNA synthesis. To understand the relationship between these two phenomena and the dynamic regulation of genes in response to upstream signals, we describe the use of live-cell RNA imaging coupled with Hi-C measurements and dissect the endogenous regulation of the estrogen-responsive TFF1 gene. Although TFF1 is highly induced, we observe short active periods and variable inactive periods ranging from minutes to days. The heterogeneity in inactive times gives rise to the widely observed “noise” in human gene expression and explains the distribution of protein levels in human tissue. We derive a mathematical model of regulation that relates transcription, chromosome structure, and the cell’s ability to sense changes in estrogen and predicts that hypervariability is largely dynamic and does not reflect a stable biological state.


RNA; chromosome; estrogen; fluorescence; heterogeneity; imaging; live-cell; single-molecule; steroid; transcription

PMID: 30554876


DOI: 10.1016/j.cell.2018.11.026

New paper on GWAS

 2018 Dec;42(8):783-795. doi: 10.1002/gepi.22161. Epub 2018 Sep 24.

The accuracy of LD Score regression as an estimator of confounding and genetic correlations in genome-wide association studies.

Author information

Department of Psychology, University of Minnesota Twin Cities, Minneapolis, Minnesota.
Mathematical Biology Section, Laboratory of Biological Modeling, NIDDK, National Institutes of Health, Bethesda, Maryland.


To infer that a single-nucleotide polymorphism (SNP) either affects a phenotype or is linkage disequilibrium with a causal site, we must have some assurance that any SNP-phenotype correlation is not the result of confounding with environmental variables that also affect the trait. In this study, we study the properties of linkage disequilibrium (LD) Score regression, a recently developed method for using summary statistics from genome-wide association studies to ensure that confounding does not inflate the number of false positives. We do not treat the effects of genetic variation as a random variable and thus are able to obtain results about the unbiasedness of this method. We demonstrate that LD Score regression can produce estimates of confounding at null SNPs that are unbiased or conservative under fairly general conditions. This robustness holds in the case of the parent genotype affecting the offspring phenotype through some environmental mechanism, despite the resulting correlation over SNPs between LD Scores and the degree of confounding. Additionally, we demonstrate that LD Score regression can produce reasonably robust estimates of the genetic correlation, even when its estimates of the genetic covariance and the two univariate heritabilities are substantially biased.


causal inference; genetic correlation; heritability; population stratification; quantitative genetics








The low carb war continues

Last month, a paper in the British Journal of Medicine on the effect of low carb diets on energy expenditure, with senior author David Ludwig, made a big splash in the popular press and also instigated a mini-Twitter war. The study, which cost somewhere in the neighborhood of 12 million dollars, addressed the general question of whether a person will burn more energy on a low carbohydrate diet compared to an average or high carb diet. In particular, the study looked at the time period after weight loss where people are susceptible to regaining weight. The argument is that it will be easier to maintain weight loss on a low carb diet since you will be burning more energy. Recent intensive studies by my colleague Kevin Hall and others have found that low carb diets had little effect if any on energy expenditure, so this paper was somewhat of a surprise and gave hope to low carb aficionados. However, Kevin found some possible flaws, which he points out in an official response to BMJ and a BioRxiv paper, which then prompted a none-too-pleased response from Ludwig, which you can follow on Twitter. The bottom line is that the low carb effect size depends on the baseline point you compare too. In the original study plan, the baseline point was chosen to be energy expenditure prior to the weight loss phase of the study. In the publication, the baseline point was changed to after the weight loss but before the weight loss maintenance phase. If the original baseline was chosen, the low carb effect is no longer significant. The authors claim that they were blinded to the data and changed the baseline for technical reasons so this did not represent a case of p-hacking where one tries multiple combinations until something significant turns up. It seems pretty clear to me that low carbs do not have much of a metabolic effect but that is not to say that low carb diets are not effective. The elephant in the room is still appetite. It is possible that you are simply less hungry on a low carb diet and thus you eat less. Also, when you eliminate a whole category of food, there is just less food to eat. That could be the biggest effect of all.

The tragedy of low probability events

We live in an age of fear and yet life (in the US at least) is the safest it has ever been. Megan McArdle blames coddling parents and the media in a Washington Post column. She argues that cars and swimming pools are much more dangerous than school shootings and kidnappings yet we mostly ignore the former and obsess about the latter. However, to me dying from an improbable event is just so much more tragic than dying from an expected one. I would be much less despondent meeting St. Peter at the Pearly Gates if I happened to expire from cancer or heart disease than if I were to be hit by an asteroid while weeding my garden. We are so scared now because we have never been safer. We would fear terrorist attacks less if they were more frequent. This is the reason that I would never want a major increase in lifespan. I most certainly would like to last long enough to see my children become independent but anything beyond that is bonus time. Nothing could be worse to me than immortality. The pain of any tragedy would be unbearable. Life would consist of an endless accumulation of sad memories. The way out is to forget but that to me is no different from death. What would be the point of living forever if you were to erase much of it. What would a life be if you forgot the people and things that you loved? To me that is no life at all.

Harvard and Asian Americans

The current trial regarding Harvard’s admissions policies seem to clearly indicate that they discriminate against Asian Americans. I had always assumed this to be the case. My take is that the problem is not so much that Harvard is non-transparent and unfair in how it selects students but rather that Harvard and the other top universities have too much influence on the rest of society. Each justice on the US Supreme Court has a degree from either Harvard or Yale. That is positively feudalistic. So here is my solution. All universities have a choice. They can 1) choose students any way they wish but they lose their tax free status or 2) retain tax exempt status but then adhere to strict non-discrimination and affirmative action rules. The top schools already have massive endowments and hurt the locales they are in by buying property and then not pay property taxes. I say let them do what they want but tax them heavily for the right to do so. The government should also not subsidize loans for students that attend such schools.

New paper on learning in spiking neural networks

Chris Kim and I recently published a paper in eLife:

Learning recurrent dynamics in spiking networks.


Spiking activity of neurons engaged in learning and performing a task show complex spatiotemporal dynamics. While the output of recurrent network models can learn to perform various tasks, the possible range of recurrent dynamics that emerge after learning remains unknown. Here we show that modifying the recurrent connectivity with a recursive least squares algorithm provides sufficient flexibility for synaptic and spiking rate dynamics of spiking networks to produce a wide range of spatiotemporal activity. We apply the training method to learn arbitrary firing patterns, stabilize irregular spiking activity in a network of excitatory and inhibitory neurons respecting Dale’s law, and reproduce the heterogeneous spiking rate patterns of cortical neurons engaged in motor planning and movement. We identify sufficient conditions for successful learning, characterize two types of learning errors, and assess the network capacity. Our findings show that synaptically-coupled recurrent spiking networks possess a vast computational capability that can support the diverse activity patterns in the brain.


The ideas that eventually led to this paper were seeded by two events. The first was about five years ago when I heard Dean Buonomano talk about his work with Rodrigo Laje on how to tame chaos in a network of rate neurons. Dean and Rodrigo expanded on the work by Larry Abbott and David Sussillo. The guiding idea from these two influential works stems from the “the echo state machine” or “reservoir computing”. Basically, this idea exploits the inherent chaotic dynamics of a recurrent neural network to project inputs onto diverse trajectories from which a simple learning rule can be deployed to extract a desired output.

To explain the details of this idea and our work, I need to go back to Minsky and Papert and their iconic 1969 book on feedforward neural networks (called perceptrons), who divided learning problems into two types.  The first type is linearly separable, which means that if you want to learn a classifier on some inputs, then a single linear plane can be drawn to separate the two input classes on the space of all inputs. The classic example is the OR function.  When given inputs (x_1,x_2) = (0,1), (1,0), (1,1), it outputs 1 and when given (0,0) it outputs 0. If we consider the inputs on the x-y plane then we can easily draw a line separating point (0,0) from the rest. The classic linearly non-separable problem is exclusive OR or XOR, where (0,0) and (1,1) map to 0, while (0,1) and (1,0) map to 1. In this case, no single straight line can separate the points. Minsky and Papert showed that a single layer perceptron, where the only thing you learn is the connection strengths from input to output, can never learn a linearly inseparable problem. Most interesting and nontrivial problems are inseparable.

Mathematically, we can write a perceptron as x_i^{\alpha+1} = \sum w_{ij}^{\alpha}f^{\alpha}(x_j^{\alpha}), where x_i^{\alpha} is the value of neuron i in layer \alpha and f is a connection or gain function. The inputs are x_i^{0} and the output are x_i^{L}. The perceptron problem is to find a set of w‘s such that the output layer gives you the right value to a task posed in the input layer, e.g. perform XOR. A single layer perceptron is then simply x_i^{1} = \sum w_{ij}^{\alpha}f^{\alpha}(x_j^{0}). Now of course we could always design f to do what you ask but since we are not training f, it needs to be general enough for all problems and is usually chosen to be a monotonic increasing function with a threshold. Minsky and Papert showed that the single layer problem is equivalent to the matrix equation w = M v and this can never solve a linearly inseparable problem, since it defines a plane. If f is a linear function then a multiple layer problem reduces to a single layer problem so what makes perceptron learning and deep learning possible is that there are multiple layers and f is a nonlinear function. Minsky and Paper also claimed that there was no efficient way to train a multi-layered network and this killed the perceptron for more than a decade until backpropagation was discovered and rediscovered in the 1980’s. Backprop rekindled a flurry of neural network activity and then died down because other machine learning methods proved better at that time. The recent ascendancy of deep learning is the third wave of perceptron interest and was spurred by the confluence of 1) more computing power via the GPU, 2) more data, and 3) finding the right parameter tweaks to make perceptron learning work much much better. Perceptrons still can’t solve everything, e.g. NP complete problems are still NP complete, they are still far from being optimal, and they do not discount a resurgence or invention of another method.

The idea of reservoir computing is to make a linearly inseparable problem separable by processing the inputs. The antecedent is the support vector machine or kernel method, which projects the data to a higher dimension such that an inseparable problem is separable. In the XOR example, if we can add a dimension and map (0,0) and (1,1) to (0,0,0) and (1,1,0) and map (1,0) and (0,1) to (1,0,1) and (0,1,1) then the problem is separable. The hard part is finding the mapping or kernels to do this. Reservoir computing uses the orbit of a chaotic system as a kernel. Chaos, by definition, causes initial conditions to diverge exponentially and by following a trajectory for as long as you want you can make as high dimensional a space as you want; in high enough dimensions all points are linearly separable if they are far enough apart. However, the defining feature of chaos is also a bug because any slight error in your input will also diverge exponentially and thus the kernel is inherently unstable. The Sussillo and Abbott breakthrough was that they showed you could have your cake and eat it too. They stabilized the chaos using feedback and/or learning while still preserving the separating property. This then allowed training of the output layer to work extremely efficiently. Laje and Bunomano took this one step further by showing that you could directly train the recurrent network to stabilize chaos. My thought at that time was why are chaotic patterns so special? Why can’t you learn any pattern?

The second pivotal event came in a conversation with the ever so insightful Kechen Zhang when I gave a talk at Hopkins. In that conversation, we discussed how perhaps it was possible that any internal neuron mechanism, such as nonlinear dendrites, could be reproduced by adding more neurons to the network and thus from an operational point of view it didn’t matter if you had the biology correct. There would always exist a recurrent network that could do your job. The problem was to find the properties that make a network “universal” in that it could reproduce the dynamics of any other network or any dynamical system. After this conversation, I was certain that this was true and began spouting this idea to anyone who would listen.

One of the people I mentioned this to was Chris Kim when he contacted me for a job in my lab in 2015. Later Chris told me that he thought my idea was crazy or impossible to prove but he took the job anyway because he wanted to be in Maryland where his family lived. So, upon his arrival in the fall of 2016, I tasked him with training a recurrent neural network to follow arbitrary patterns. I also told him that we should do it on a network of spiking neurons. I thought that doing this on a set of rate neurons would be too easy or already done so we should move to spiking neurons. Michael Buice and I had just recently published our paper on computing finite size corrections to a spiking network of coupled theta neurons with linear synapses. Since we had good control of the dynamics of this network, I thought it would be the ideal system. The network has the form

\dot\theta_i = f(\theta_i, I_i, u_i)

\tau_s \dot u_i= - u_i+ 2 \sum_j w_{ij}\delta(\theta_j-\pi)

Whenever neuron j crosses the angle \pi it gives an impulse to neuron i with weight scaled by w_{ij}, which can be positive or negative. The idea is to train the synaptic drive u_i(t) or the firing rate of neuron i to follow an arbitrary temporal pattern. Despite his initial skepticism, Chris actually got this to work in less than six months. It took us another year or so to understand how and why.

In our paper, we show that if the synapses are fast enough, i.e. \tau_s is small enough, and the patterns are diverse enough, then any set of patterns can be learned. The reason, which is explained in mathematical detail in the paper, is that if the synapses are fast enough, then the synaptic drive acts like a quasi-static function of the inputs and thus the spiking problem reduces to the rate problem

\tau_s \dot u_i= - u_i+ \sum_j w_{ij}g(u_j)

where g is the frequency-input curve of a theta neuron. Then the problem is about satisfying the synaptic drive equation, which given the linearity in the weights, boils down to whether \tau_s \dot u_i + u_i is in the space spanned by \sum w_{ij} g(u_j),  which we show is always possible as long as the desired patterns imposed on u_i(t) are uncorrelated or linearly independent enough. However, there is a limit to how long the patterns can be, which is governed by the number of entries in w_{ij}, which is limited by the number of neurons. The diversity of patterns limitation can also be circumvented by adding auxiliary neurons. If you wanted some number of neurons to do the same thing, you just need to include a lot of other neurons that do different things. A similar argument can be applied to the time averaged firing rate (on any time window) of a given neuron. I now think we have shown that a recurrent network of really simple spiking neurons is dynamically universal. All you need are lots of fast neurons.


Addendum: The dates of events may not all be correct. I think my conversation with Kechen came before Dean’s paper but my recollection is foggy. Memories are not reliable.