Supplementary information for:

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1 Supplementary information for: Synaptic polarity of the interneuron circuit controlling C. elegans locomotion Franciszek Rakowski a, Jagan Srinivasan b, Paul W. Sternberg b, and Jan Karbowski c,d a Interdisciplinary Center for Mathematical and Computational Modeling, University of Warsaw, Warsaw, Poland b California Institute of Technology, Division of Biology, Pasadena, CA 91125, USA c Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland d Institute of Biocybernetics and Biomedical Engineering, Polish Academy of Sciences, Warsaw, Poland 1

2 Activity equations for interneurons. The dynamics of the interneurons in the locomotory circuit are based on the derived Eq. (4) in the main text. They are represented by a following set of differential equations: τ dav B = AV B + ǫ ASH w AV B,ASH H(ASH) + ǫ PV C w AV B,PV C H(PV C) +ǫ AV A w AV B,AV A H(AV A) + ǫ AV D w AV B,AV D H(AV D) +ǫ DV A w AV B,DV A H(DV A) + ǫ 2 AV B ǫ2 DV A g AV B,DV A(DV A AV B) +ǫ 2 AV B g AV B,F(E f AV B) + ǫ 2 AV B g AV B,B(E b AV B) + X AV B, (1) τ dpv C = PV C + ǫ AV A w PV C,AV A H(AV A) + ǫ DV A w PV C,DV A H(DV A) +ǫ AV D w PV C,AV D H(AV D) + ǫ AV E w PV C,AV E H(AV E) +w PV C,F H(E f ) + w PV C,B H(E b ) +ǫ 2 PV C ǫ2 AV A g PV C,AV A(AV A PV C) + ǫ 2 PV C ǫ2 DV A g PV C,DV A(DV A PV C) +ǫ 2 PV C g PV C,F(E f PV C) + ǫ 2 PV C g PV C,B(E b PV C) + X PV C, (2) 2

3 τ ddv A = DV A + ǫ PV C w DV A,PV C H(PV C) + w DV A,F H(E f ) +ǫ 2 DV A ǫ2 AV B g DV A,AV B(AV B DV A) + ǫ 2 DV A ǫ2 PV C g DV A,PV C(PV C DV A) +ǫ 2 DV Ag DV A,F (E f DV A) + X DV A, (3) τ dav A = AV A + ǫ ASH w AV A,ASH H(ASH) + ǫ AV B w AV A,AV B H(AV B) +ǫ PV C w AV A,PV C H(PV C) + ǫ AV D w AV A,AV D H(AV D) +ǫ AV E w AV A,AV E H(AV E) + ǫ DV A w AV A,DV A H(DV A) +w AV A,B H(E b ) + ǫ 2 AV A ǫ2 PV C g AV A,PV C(PV C AV A) +ǫ 2 AV A g AV A,B(E b AV A) + ǫ 2 AV A g AV A,F(E f AV A) + X AV A, (4) τ dav D = AV D + ǫ AV B w AV D,AV B H(AV B) + ǫ PV C w AV D,PV C H(PV C) +ǫ ASH w AV D,ASH H(ASH) + ǫ AV A w AV D,AV A H(AV A) +ǫ AV E w AV D,AV E H(AV E) + w AV D,B H(E b ) + X AV D, (5) 3

4 τ dav E = AV E + ǫ AV B w AV E,AV B H(AV B) + ǫ PV C w AV E,PV C H(PV C) +ǫ DV A w AV E,DV A H(DV A) + ǫ ASH w AV E,ASH H(ASH) +ǫ AV A w AV E,AV A H(AV A) + X AV E, (6) where AV B, PV C, DV A, AV A, AV D, AV E are the relative activities of the corresponding interneurons with respect to their resting values. The parameters w ij are synaptic strengths, and g ij are gap-junction (electrical) couplings, where the subscripts i and j denote the above interneurons. The symbol ǫ i denotes synaptic polarity of the neuron i, and it assumes value 1 (if the neuron is excitatory), value 1 (if inhibitory), or 0 (if the neuron is absent because of the ablation). Note that gap junction couplings contain the ǫ 2 i factors, which are either 1 (if the neuron i is present), or 0 (if it is removed from the network). The parameter X i describes a constant in time input coming from the upstream neurons to the interneuron i. For all interneurons except PVC this input comes from the head neurons. It is represented by X i = x 0 +σz i, where x 0 = 2.0 mv, σ is a variable parameter, and z i is either 0 (weak input) or 1 (strong input). The parameter z i, similar to ǫ i, is unknown. We want to find both of them for each interneuron. The above pre-motor interneurons make synaptic and gap junction connections with downstream excitatory motor neurons. Two separate groups of these motor neurons generating forward and backward motion, called B and A respectively, directly connect locomotory muscles. The activities of excitatory motor neurons are given by: 4

5 τ de f = E f + ǫ PV C w F,PV C H(PV C) + ǫ DV A w F,DV A H(DV A) +ǫ AV A w F,AV A H(AV A) + ǫ AV B w F,AV B H(AV B) +ǫ AV D w F,AV D H(AV D) + ǫ AV E w F,AV E H(AV E) +ǫ 2 AV B g F,AV B(AV B E f ) + ǫ 2 AV A g F,AV A(AV A E f ) +ǫ 2 PV C g F,PV C(PV C E f ) + ǫ 2 DV A g F,DV A(DV A E f ), (7) and τ de b = E b + ǫ AV A w B,AV A H(AV A) + ǫ AV D w B,AV D H(AV D) +ǫ AV E w B,AV E H(AV E) + ǫ AV B w B,AV B H(AV B) +ǫ PV C w B,PV C H(PV C) + ǫ DV A w B,DV A H(DV A) +ǫ 2 AV Ag B,AV A (AV A E b ) + ǫ 2 AV Bg B,AV B (AV B E b ) + ǫ 2 PV Cg B,PV C (PV C E b ), (8) where E f and E b are the total relative activities of forward (type B) and backward (type A) motor neurons, respectively, measured from their resting voltages. Three state model of C. elegans locomotion. We assume that C. elegans locomotion can be described approximately as a three state model. These three states correspond to forward movement, backward movement, 5

6 and stopped time (no motion). The average times the worm spends in each state are denoted as T f, T b, and T s, respectively. With each state we associate probabilities of occurance, as is explained below. The probability of forward motion P f can be written as P f = Z exp[(e f E b )/η 0 ], (9) where is some activity threshold for locomotion initiation, the parameter η 0 characterizes the level of noise in the system, and Z is the normalization factor. By symmetry considerations, we can write the probability for backward motion P b as P b = Z exp[(e b E f )/η 0 ]. (10) The choice of the exponential function in P f and P b is motivated by two major arguments. First, with the exponentials both P f and P b are always increasing and positive functions of the arguments E f E b and E b E f for the whole range of their variabilities (from to + ), which is generally not the case for other simple choices, in particular polynomials. For example, the choice P f (E f E b ) n, with n an even integer, is not satisfactory because the probability P f is a non-monotonic function of its argument (it has a minimum for E f = E b + ). Similarly, if the exponent n is an odd integer, then P f becomes negative for E f < E b +, which is clearly wrong. A second argument in favor of the exponential function in the probabilities is the fact that many 6

7 phenomena occurring in nature have a similar type of dependence. The probability that the nematode does not move, i.e. it is in the stopped state, is ( ) Ef E b P s = Z S s, (11) where S s (x) is some unknown function that should have the following properties. For x 1, i.e. when E f E b is much smaller than the motion activation threshold, the function S s (x) 1. For x 1, i.e. when either E f or E b dominates, the worm should not be motionless, which corresponds to S s (x) 0. Additionally, due to symmetry one should have S s ( x) = S s (x). The form of the S s (x) function is however irrelevant for the kind of computations made in this paper (see below). From the normalization condition for probabilities, P f + P b + P s = 1, we can determine Z, which allows us to find explicit forms for the state probabilities. The latter read: P f = e (E f E b )/η 0 e (E f E b )/η 0 + e (E b E f )/η 0 + Ss (12) P b = e (E b E f )/η 0 e (E f E b )/η 0 + e (E b E f )/η 0 + Ss (13) 7

8 P s = S s e (E f E b )/η 0 + e (E b E f )/η 0 + Ss (14) In a case when the activity of forward motor neurons dominates over the rest (E f E b + ), we have P f 1, P b 0, and P s 0. For a balanced activation of forward and backward motor neurons, i.e. when E f E b, we have S s 1, which leads to P f P b 1, and P s 1. On the other hand the probabilities P f, P b, and P s are related to the times (T f, T b, T s ) the worm spends in the corresponding states. The average probability that C. elegans is in the forward state is P f = T f /(T f +T b +T s ). Similarly, P b = T b /(T f +T b +T s ), and P s = T s /(T f + T b + T s ). Thus, the ratio T f /T b is equal to P f /P b. Combining the above equations, we obtain T f /T b = exp [(E f E b )/η], (15) where η = η 0 /2. Note that the ratio of times associated with forward and backward locomotion neither depends on the activation threshold nor on the form of the S s (x) function. The ratio of times depends only on the difference in the activation of complementary motor neurons and the level of noise in the system. It is interesting to stress that the quantity of empirical interest, i.e. T f /(T f +T b ) is equal to [1 + exp((e b E f )/η)] 1. The latter expression is known as a sigmoidal logistic function, and serves as a transfer 8

9 function from neural activities to behavioral output. In order to examine the robustness of our results, we also investigated another choice for T f /T b as a function of E f E b, different from that given by Eq. (8) in the main text. Specifically, we tried the following function: T f /T b = ln(1 + e (E f E b )/η )/ ln2. (16) Note that for E b E f, Eqs. 15 and 16 both yield T f /T b that are proportional to each other. For E f E b they behave differently, i.e. the ratio T f /T b increases with E f E b much faster in Eq. (15) (exponential) than in Eq. (16) (linear). We found that the winning combinations had essentially the same ED values for both choices of T f /T b, which implies that Eqs. (15) and (16) yield equivalent results for the best configurations. This means that our results are not sensitive to the precise form of the transfer function between neural activities and locomotory output. 9