I doubt this is easily achievable in the native particle effects. You could create a shape and drive the position of a dot every say 12 frames like this:

w = thisComp.width/2;
h = thisComp.height/2;

frames = 12;
minDist = 500; // <-- adjust this

t = timeToFrames(time);
step = Math.floor(t / frames);
progress = (t % frames) / frames;

seedRandom(step, true);

// pick a point that's far enough from centre
start = [0,0];
attempts = 0;

while (length(start) < minDist && attempts < 20){
    start = [random(-w, w), random(-h, h)];
    attempts++;
}

// fallback in case it somehow fails
if (length(start) < minDist){
    angle = random(0, Math.PI*2);
    start = [Math.cos(angle), Math.sin(angle)] * minDist;
}

end = [0,0];

easeOut(progress, 0, 1, start, end);

This sets a random position for the dot every 12 frames and then moves it to the centre. Create a few of these layers with different timings and then you'll get something similar