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Bayesian solution to inverse problem (ill-posed)

  1. May 17, 2013 #1
    I posted it in math, But I think maybe it is more physics then math...

    Hello,
    I'm trying to understand the algorithm(‘‘juggling search’’) used in the following article :

    http://www.cns.atr.jp/~kawato/Ppdf/1...00105-main.pdf [Broken]


    In short they have a model of IO cells that are connected via gap junctions. for each different[0:0.025:0.2] gc( gap junction conductance ) and gi ( inhibitory synapse conductance) they simulate firing dynamics.
    They also have experimental data of IO cells from control (CON) and CBX ( gap junction blocker) and PIX ( inhibitory junction blocker) , they fractionated the EXP and SIM data to spatiotemporal segments (short time segments and small neuronal subgroups), and searched for the SIM data of the best fit (i.e., the one with the minimum error from the EXP one) segment by segment. (each EXP and SIM segment was analysed to feature vectors, and then to PCA space)

    My problem is in understanding the following algorithm:

    "The gc and gi for the SIM spike segments that showed the closest match (the minimum PCA errors) to the EXP spike segments were searched in the PCA space as the solution to the inverse problem to estimate gc and gi
    from the firing dynamics. However, a global search of the best fit with no constraint failed to resolve the inverse
    problem, due to the ill-posed nature of the inverse problem, because the major determinant of the IO firing dynamics was the gc/gi ratio rather than either individual value (Katori et al., 2010; Onizuka, 2009). This issue was resolved by a ‘‘juggling algorithm’’, in which the match between the SIM and EXP spike segments was searched under the constraint that gc and gi remain unchanged by PIX and CBX administration, respectively. This constraint was
    based the experimental facts that PIX’s action should reduce gi but have no effect on gc , whereas the converse is true for CBX. Therefore, the estimated distributions of gc should overlap each other between the CON and PIX conditions, and similarly those for gi in the CON and CBX conditions should overlap. The juggling algorithm consisted of four steps. First, a global search with no constraint was conducted to find gi and gc for CON spike segments sampled in both PIX and CBX experiments. Second, a constrained search was conducted to find the partner
    gi that paired with the ensemble gc for the CON spike segments determined by the global search. Third, a similar constrained search was conducted to find the partner gc paired with the ensemble gi for the CON and PIX spike segments in the PIX experiments, determined by the second step search. Fourth, another constrained search was conducted to find the partner gi paired with the ensemble gc for the CON and PIX spike segments in the PIX experiments, determined by the third step search. The loop of steps 2–4 was repeated until the ensemble of gc for PIX experiments matched with that for CON experiments and so was repeated until the ensemble of gi for CBX experiments matched with that of CON experiments."

    I don't exactly manage to understand what is a constrain search.. what are the equation of this kind of search? What exactly does it means?


    in the methods they also write :
    "Our study to estimate the synaptic conductance of the IO circuitry from neuronal firing using IO network simulation may be regarded as a Bayesian solution to the inverse problem in the following sense. According to Bayes’ rule, the posterior probability of the synaptic conductance is given as the product of the likelihood and the prior probability of the synaptic conductance. To obtain a value for the likelihood, the network simulation played the role of the forward model to predict firing dynamics from a given set of synaptic conductances. The match probability between the
    SIM and EXP data in our analysis then corresponds to the likelihood of a specific EXP data given a specific parameter set. Strictly, the likelihood was approximated by a Gibbs distribution with a lowtemperature limit, while the energy term is the squared error between the EXP and SIM firing patterns. Global search assumed the prior probability to be a non-informative uniform distribution, and a ‘‘juggling search’’ that will be introduced later utilized the
    posterior probability determined by the global search as the prior probability. The inverse problem to estimate gc and gi from IO firing dynamics was resolved in four steps. First, SIM spike data were generated with step-wise variations of gi and gc by an IO network model. Second, the complicated firing of the EXP and SIM spike data that vary in space (from neuron to neuron) and time were fractionated into spatiotemporal segments (short spike segments
    for small groups of neurons) and evaluated in a segment-wise fashion using firing feature vectors (FVs), and the FVs of the spike data segments were contracted onto two-dimensional PCA space. Thus the firing dynamics of the EXP and SIM spike data were mapped onto the PCA space for each spatiotemporal data segment. Third, the gc and gi
    for the SIM spike segments that showed the closest match in the PCA space (i.e., the minimum PCA errors defined as Euclidean distance) to the EXP ones were determined as the solutions to the inverse problem. Fourth, there was an ill-posed problem that the firing dynamics mainly depend on the ratio of gc to gi rather than individuals (Katori, Lang, Onizuka, Kawato, & Aihara, 2010; Onizuka, 2009). This issue was resolved by a ‘‘juggling algorithm’’ in which the match between the SIM and EXP spike segments was searched under the constraint that gc and gi remain unaffected by PIX and CBX administration, respectively. In other words, gc and gi should agree with each other between CON and PIX conditions and between CON and CBX conditions, respectively.

    I thank you.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
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