The Model

This learning algorithm was quantitatively assessed in [AWP16], where the memory ca-pacity and the noise tolerance were tested in the most simple neuron model presented here, the integrate-and-fire neuron.

7.2.1.2 The conductance-based Integrate-and-Fire Neuron

The simple model above suffers from the fact that MPDP is agnostic to the type of synapse.

In principle, MPDP can turn excitatory synapses into inhibitory ones by changing the sign of any weightwi. Since this is a violation of Dale’s law, a more biologically realistic scenario involving MPDP is presented.

The presynaptic population is split intoN_ex excitatory andN_in inhibitory neurons. The postsynaptic neuron is modelled as a conductance based LIF neuron governed by

Cm

dV

dt =−g_L(V −VL)−(g_sl+g_f)(V −V_h)−gex(V −Vex)−gin(V −V in) , (7.3) where V denotes the membrane potential, C_m = 0.16µF the membrane capacitance, V_L = −70mV the resting potential, g_L = 20 the leak conductance, V_i = −75mV and Vex = 0 the reversal potential of inhibition and excitation, respectively and gin and gex

their respective conductances. The spike after-hyperpolarisation is modelled to be biphasic consisting of a fast and a slow part, described by conductances g_f and g_sl that keep the membrane potential close to the hyperpolarisation potentialVh=Vi. When the membrane potential surpasses the spiking threshold V_thr = −50 at time t_post, a spike is registered and the membrane potential is reset toV_reset =V_h. All conductances are modelled as step and decay functions. The reset conductances are given by

τ_f,slg˙_f,sl=−g_f,sl+ ∆g_f,sl∑

tpost

δ(t−tpost) , (7.4)

where ∆g_sl = 5 resp. ∆g_sl = 1000 is the increase of the fast and slow conductance at the time of each postsynaptic spike. They decay back with time constants τ_f = τ_s <

τ_sl = C_m/g_L. The input conductances g_ex and g_in are step and decay functions as well, which are increased bywi when presynaptic neuronispikes and decay with time constant τ_s= 2ms. w_i denotes the strength of synapsei.

7.2.1.3 The Hodgkin-Huxley-type Neuron

To test whether the learning mechanism suggested below is also capable of learning spike associations for neuron models that generate spikes, it is tested on a Hodgkin-Huxley type neuron (see section 3.1.2.3).

Inhibitory and excitatory inputs remain separated into two input populations. They pro-vide input into the output neuron with membrane potentialV(t) given by

CmV˙ =−g_L(V −VL)−gKn⁴(V−VK)−gN am³h(V −VN a)−gex(V−Vex)−gin(V −V in) (7.5) where V_L = −65mV is the leak potential, g_L = 0.1mS/cm is the leak conductance, gex resp. gin are the conductance governing excitatory resp. inhibitory input from the input populations and Vex = 0mV resp. Vin = −75mV are their reversal potentials.

V_{N a} = 55mV is the reversal potential of sodium,V_K =−90mV is the reversal potential of potassium,Cm= 1µF/cm²is the membrane capacitance,gK = 9mS/cm² is the maximum potassium conductance andgN a= 35mS/cm² is the maximum sodium conductance.

The conductance variables n(t, V), m(t, V) and h(t, V) are time and voltage dependent

and take values between 0 and 1. They are given by

˙

n = αn(V)(1−n)−βn(V)n (7.6)

˙

m = αm(V)(1−m)−βm(V)m (7.7)

h˙ = α_h(V)(1−h)−β_h(V)h (7.8) where

α_n(V) = (−0.01(V(t) + 34))

(exp(−0.1(V + 34))−1) (7.9)

β_n(V) = 0.071025 exp(−(V(t) + 75)

500 ) (7.10)

αm(V) = (−0.1(V + 35))

(exp(−0.1(V + 35))−1) (7.11)

β_m(V) = 4 exp(−V + 68

18) (7.12)

α_h(V) = 0.07 exp(−(V + 66)

20 ) (7.13)

βh(V) = 1

exp(−0.1(V + 28)) + 1 (7.14)

To facilitate reading, units are dropped; voltages are in mV, time is inms. Parameters are adapted from experimental values to yield a strong hyperpolarisation.

7.2.2 Learning Rule

The plasticity rule is derived from the demand of a balanced membrane potential: The neuron should neither be hyperpolarized nor too strongly depolarized. This is a sensible demand, because it holds the neuron at a sensitive working point and keeps metabolic costs down. To that end, two thresholds are introduced, ϑ_P < ϑ_D < V_thr, between which the membrane potential is bounded. The weight change is chosen such that, whenever ϑ_D = 10mV is surpassed, all weights that contribute to the rise of the membrane potential are depressed, weighted by their respective influence given by the PSP-kernelε. Whenever the membrane potential drops belowϑ_P =V_L, all synapses that contribute to that downward deflection are potentiated, such that for a repetition of the pattern the membrane potential is deflected to stay within bounds. Additionally, in some cases (a = 1) the weights are bounded to stay below a maximum weightw_max, symbolizing a maximal synaptic strength, while in other cases the weights are not limited (a= 0). Limiting the maximum weights is advantageous for stability in the case of synapses that can change signs and a real teacher input. The weight change is then given by

˙

w_i=η(w_max− |w_i|)^a(

−γ[V(t)−ϑ_D]₊+ [ϑ_P −V(t)]^b₊) ∑

k

ε(

t−t^k_i −t_delay)

. (7.15) The parameterbdefines, whether the contribution of the potentiation term scales linearly (b = 1) or quadratically (b = 2) with the distance of the membrane potential from the plasticity threshold. γ is a factor that scales inhibition to excitation, which needs to be adapted to the neuron model in question. γ = 650 for the simple integrate-and-fire neuron

and γ = 150 for the conductance-based integrate-and-fire neuron. For the Hodgekin-Huxley type neuron, γ = 20.

Obviously, the PSP-kernel used in the learning rule only has a very direct interpretation in the case of simple linear integrate-and-fire neurons. However, in the learning rule, this term only serves to estimate the extent to which the postsynaptic membrane potential depends on the input of one particular neuron. Hence, in the non-linear neuron models a pseudo-PSP is tracked and used instead (with τm = Cm/gL). Furthermore, for the conductance-based integrate-and-fire neuron, the upper threshold is chosen to lie between resting potential and firing threshold: ϑD =−53mV.

For the Hodgkin-Huxley type neuron, there is no spiking threshold. To avoid an overly strong influence of the very high membrane potential during the spike, the LTD part of the learning rule is clipped to the respective constant value at ϑ^up_D =−55mV. The plasticity thresholds are chosen asϑD =−58mV and ϑP =−64mV.

For the inhibitory synapses coming into play for conductance-based neurons between the inhibitory presynaptic neurons and the output neurons, the learning rule is adapted such that the net effect of learning on the membrane potential is preserved. To that end, the same learning rule as for the excitatory synapses is applied, just with the opposite sign:

˙

w^inh =−w˙ (7.16)

7.2.2.1 Chronotron Setup

Consider a feed-forward network consisting of N = 200 presynaptic neurons and one postsynaptic neuron. For illustration purposes, each input neuron spikes once in each of the five patterns used during training. To train the output neuron to spike at a specific time in response to each input pattern, a single spike is induced at a fixed timetpost= 100ms by a supplementary external (teacher) input

Iext=cexp (

−t−tpost

τ_s )

Θ(t−tpost). (7.17)

c is the amplitude of the teacher input. The shape of the current is chosen to mimic a synaptic input and induce a PSP-like voltage perturbance (see eq. (9.5)).

While in the case of the simple integrate-and-fire neuron synapses change signs, for the conductance-based integrate-and-fire neuron and the Hodgekin-Huxley type neuron positive and negative inputs need to be separated. To that end, the output neuron receives inhibitory input fromN_in = 200 inhibitory presynaptic neurons and excitatory input from Nex = 200 excitatory presynaptic neurons. An input pattern for learning thus consists of a set of one excitatory and one inhibitory input pattern. In each pattern, each input neuron spikes once.

During learning, the patterns are presented to the neuron concurrent with the teacher input. Weights are updated after all patterns were presented (batch mode). For recall, just the input patterns are presented and if learning was successful, the output neuron will generate a spike close to the time of the (now absent) teacher spike.

Figure 7.1: (a) The model network has a simple feed-forward structure. The top picture shows three pre- and one postsynaptic neurons, connected by synapses. Line width in this example corresponds to synaptic strength. The bottom picture shows the postsynaptic membrane potential in response to the input. (b) Illustration of Anti-Hebbian Membrane Potential Dependent Plasticity (MPDP). A LIF neuron is presented twice with the same presynaptic input pattern. Excitation never exceeds Vthr. MPDP changes synapses to counteract hyperpolarization and depolarization occuring in the first presentation (blue trace), reducing (arrows) them on the sec-ond presentation (green trace). (c) Homeostatic MPDP on inhibitory synapses is compatible with STDP as found in experiments. Plasticity is tested for different temporal distances between pre- and postsynaptic spiking; the resulting spike tim-ing characteristic is in agreement with experimental data on STDP of inhibitory synapses [HNA06].